REVISE© AND ENLARGED. 



TREATISE 



ON 



SURVEYING AND NAVIGATION 



UNITING 



THE THEORETICAL, PRACTICAL, AND 



EDUCATIONAL FEATURES OF THESE SUBJECTS, 



Br HORATIO N. ROBINSON, A. M., 

FORMERLY PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVY; AUTHOR OF 
MATHEMATICAL, PHILOSOPHICAL, AND ASTRONOMICAL WORKS. 



FOURTH STANDARD EDITION. 



CINCINNATI: 
JACOB ERNST, No. 112 MAIN STREET. 

TOLEDO: 
ANDERSON, BURR, & CO. 

1 857. 




rh 






Entered, according to act of Congress, in the year 1S52, 

By H. N. ROBINSON, 

in the Clerk's Office of the District Court for the Northern District of New York. 



Entered according to act of Congress, in the year 1857, 

By EL N. ROBINSON, 

in the Clerk's Office of the District Court for the Northern District of New York. 



.4. C. JAMES, STEREOTYFEB, 

167 WALNUT ST , CINCINNATI 



L~Z& 



->c?/t> 



PREFACE. 



This book is more than its title page proclaims it to be : it is the 
practical application of the Mathematical Sciences to Mensuration, to 
Land Surveying, to Leveling, and to Navigation. 

Nor is the work merely practical. Elementary principles are here 
and there brought before the mind in a new light ; and original investi- 
gations will be found in many parts of the work. To show the reader 
how a thing is to be done, is but a small part of the object sought to be 
obtained : the great stress is put upon the reasons for so doing, which 
gives true discipline to the mind, and adds greatly to the educational 
value of any book. 

We have illustrated the subject of logarithms, and their practical 
uses, the same in this book as is common to be found in other books, and 
this is sufficient for the common pupil, or the ordinary practical man, 
whether surveyor or navigator ; but in addition to this, we have carried 
logarithms much further in this work than in any I have seen. I do 
not mean by this that we have more voluminous tables than others. 
Such is not the fact. 

Voluminous tables are not necessary for those who really understand 
the nature of logarithms, and such are mainly intended for those who 
are not expected to understand principles. To give a more practical 
illustration of logarithms, and to suggest artifices in using logarithms 
generally, we have given Table III and its auxiliaries, on page 70 of 
tables, showing logarithms to twelve places of decimals, a degree of 
accuracy which practice never demands. By the help of this table 
combined with a true knowledge of the subject, the logarithm of any 
number may be readily found true to ten places of decimals, or, conversely 
the number corresponding to any given logarithm may be found to 
almost any degree of accuracy. 

Our Traverse Table is not so full as in some other books, but it 



v PREFACE. 

is full enough to answer every purpose ; and latitude and departure, 
corresponding to any course and distance, can be found by it, provided 
the operator's good judgment is awake. Indeed a contracted table, in 
an educational point of view, is better than a full one ; for the former 
calls forth and cultivates tact in the student, but the latter is best for 
the unanimated plodder. 

In running lines, and computing the areas of surveys, we have 
endeavored to present the subject in such a manner that the reader must 
constantly keep Elementary Geometry in view, and the whole is so 
clear and simple, that many will think it unworthy of the rank that it 
seems to hold in the public estimation, but there are other reasons for 
this. 

The chapter on surveys and surveyors will be found to be a little 
peculiar, but the information there given, will be highly useful to all 
those who are inclined to look upon a survey as a mathematical problem 
only. 

On the compass, and the declination of the needle, we have been 
very full : the subject embraces meridians and astronomical lines drawn 
on the earth. 

The manner in which we should proceed to make a survey, provided 
no such instrument as the compass existed, and there were no such 
thing as a magnetic needle, is taken up and illustrated in this work. 

The subject of dividing lands is fully discussed and illustrated, and 
if any one has occasion to complain of mathematical abstrusity in this 
work, it will be found in this connection ; yet there is nothing here 
above elementary algebra and geometry. 

The method of taking levels and making a profile of the vertical 
section of a line for rail roads, is set forth in this work. 'The profile 
shows the necessary excavation or embankment, which it is necessary 
to cut down or build up at any particular point, to conform to any 
proposed grade that may be contemplated. 

To determine the elevation of any place above the level of the sea, 
by means of the barometer, has been, and now is, a very interesting pro- 
blem to all philosophical students, yet very few of them have been able 
to comprehend it beyond its first great principle, the variation of atmos- 
pheric pressure. To trace, or rather to discover the mathematical law 
which connects the elevation of any locality with the mean hight of the 
barometer at the same place, has been an obscure problem, and we have 
taken hold of it with a determination to break open some avenue of 
light (if such were possible) by which the simplicity of the problem 
might be brought to the comprehension of the every-day mathematical 
Btudent, and we believe that we have succeeded in the undertaking. 



PREFACE. v 

The part on Navigation, might be regarded, at first view, an abridg- 
ment of that subject, and in one sense it is, for we have studied to be 
as brief as possible, but we would never let brevity stand in the place of 
perspicuity ; and however it may appear, we have given all the 
mathematical essentials of the subject, and whoever acquires what is 
here given, will find very little necessity of looking elsewhere for the 
continuation of the study, unless it is for sea terms and seamanship ; 
but these have nothing to do with Navigation as a science. Our method 
of working lunars is more brief than any other, where auxilliary tables 
and methods of approximation are not resorted to, but to attain this 
brevity, we have been compelled to use Natural Sines in part of the 
operation ; but on the other hand, this should be no objection, for it 
gives us a clearer view of the unity and harmony of the mathematical 



PREFACE TO THE FOURTH EDITION. 



For reasons which we have not here the space to explain, we have thought 
best to remove the matter between the 32d and 43d pages of the former edi- 
tions, to the last chapter in the book: to fill up that space with more simple 
and more practical matter, and to enrich the volume with additional pages 
containing very choice miscellaneous matter. We are induced to make these 
improvements, by the strong conviction that this work contains all the essen- 
tials of popularity and permanency. 

Objections have been made to the brevity of our traverse table, and at one 
time we thought of enlarging it — but on further reflection, we concluded 
that this was a mere objection, given out for the want of a better one. This 
is designed as an educational volume, and a properly educated person does 
not require a voluminous traverse table. Such tables are mostly intended 
for those who do not pretend to uuderstand them, and they are really required 
only for about one in a thousand of those who study this subject. The sur- 
veyor who is in constant practice, and such persons, have tables separate 
from all other matter, in such a form as to roll up and carry in the pocket. 



CONTENTS 



INTRODUCTION. 

CHAPTER I. 

Page 

Introduction 9 

Construction of Geometrical Problems, with the use of instru- 
ments 12 

CHAPTER II. 

Logarithms 22 

Application of Logarithms to Multiplication, Division, and the 

extraction of Roots 29 — 32 

Artifices in the use of Logarithms (Art. 12) 32 — 34 

Logarithms without the use of a table 34—43 

CHAPTER III. 

Plane Trigonometry 44 

Explanation of Tables .. 58 

Oblique-angled Plane Trigonometry 64 

SURVEYING. 

Introductory Remarks 70 

Finding Areas in general ............... 70 — 79 

Mensuration of Solids 79 

CHAPTER I. 

Mensuration of Lands 80 

To measure a Line 81 

Surveyor's Compass 82 — 85 

Vernier Scales in general 85 — 86 

CHAPTER II. 

Latitude and Departure 87 

Taking angles by the Compass 89 

(vi) 



CONTENTS. vii 

Page. 

To close a Survey 89 

To find the true from a random Line 90 

Computation of Areas by trapezoids 91 — 103 

CHAPTER III. 

To find Meridian Lines 103 

Variation of the Compass 104 — 112 

Practical Difficulties 112 

CHAPTER IV. 

To Survey without a Compass 113 

The Circumferentor 114 

CHAPTER V. 

Original and subsequent Surveys 116 

Difficulties and Duties of a Surveyor 118 

United States' Land 121 

CHAPTER VI. 

Very irregular Figures 123 

Division of Lands — a variety of Problems 124 — 145 

CHAPTER VII. 

Triangular Surveying 146 

The Plane Table— its uses &c 147—152 

Marine Surveying 152 

Piloting Ships 154 

CHAPTER VIII. 

Leveling 155 

Description of the Level 158 

Adjusting the Level 159 — 161 

Keeping Book 163 

Contour of Ground 165 

Elevation determined by atmospheric Pressure, as indicated by 

the Barometer 166 — 174 

NAVIGATION. 

CHAPTER I. 

Introduction 175 

The Log-line 176 

The Mariner's Compass 177 



viii CONTENTS. 

CHAPTER II. 

Paoe. 
Plane Sailing 180—186 

Middle Latitude Sailing 186 

Traverse Sailing 186 

Sailing in Currents 191 

CHAPTER III. 

Mercator's Chart — its construction 194 — 196 

Mercator's Sailing 196 

CELESTIAL OBSERVATIONS. 
CHAPTER I. 

Definition of Terms 198—199 

Quadrant and Sextant 202 

Construction of the Sextant 204 

The adjustments of the Sextant 205 

To take an Altitude of the Sun at sea 206 

To find the Latitude by the Sun or Stars 207 

To find the Latitude by the meridian Altitude of the Moon . 209 — 211 

CHAPTER II. 

A perfect Time-piece 213 

Local Time— Rule to find it 216 

Longitude by Chronometer 217 

CHAPTER III. 
Lunar Observations 220 

Formulae, for clearing the observed Distance from the effects of 

Parallax and Refraction 223 

Examples for working Lunars . 224 — 227 

APPENDIX. 

Artifices to be resorted to in difficult circumstances .... 228 

Results corresponding to assumed errors of observation 231 

Application of the differential calculus, to find results corres- 
ponding to eiTors of observation 232 

Application of the same, to clearing lunar distances 233 

Change of the formulas, to avoid the use of Natural sines and Nat- 
ural cosines 235 

Artifices, in the use and computation of logarithms 239 — 246 



INTRODUCTION. 



CHAPTER I. 

Mensuration, Surveying, and Navigation, are but branches of the 
same science, and should be regarded as the application of geometry 
and trigonometry, and in this light we shall present them to our 
readers. 

In this volume we shall not demonstrate geometrical truths unless 
we wish to present them in some new form, or unless the demon- 
stration is not readily to be found in the proper places, in the elemen- 
tary books. 

It is expected that all readers of a work of this kind, have 
previously made themselves more or less acquainted with Algebra 
and Geometry, and where this is the case the reader will have no 
difficulty ; and readers who are not thus prepared should be careful 
not to charge imaginary defects to the book : in no work of this 
kind would it be proper to demonstrate every elementary principle. 
These remarks apply only to the educational character of the book. 

Preparatory to a course of practical mathematics, it is proper to 
give such descriptions of the instruments to be used as will enable 
the operator to understand their use. But some of these instru- 
ments can never be understood from a book, it must be from the 
instrument itself ; we might as well attempt to give a person an idea 
of color by the means of language, as to give a person a correct idea 
of the sextant and theodolite by a mere book description. It is true 
we can do something by drawings and descriptions, and that some- 
thing we intend to do. 

To represent plane surfaces and tracts of land on paper, no other 
instruments are necessary than the scale, and dividers, and a pro- 

(9) 



10 SURVEYING. 

tractor to measure angles. In fact, every thing can be done with 
the scale and dividers only — other instruments, as the protractor, 
sector, and parallel rulers,only add to our convenience ; at the same 
time they could be dispensed with. 

THE PLANE S CALE. 

The plane scale, or the plane diagonal scale of equal parts, as here 
represented, is the most common and useful of all the instruments 
used in drawing. It is also a ruler, and if wide and well made will 
serve as a square also, by which right angles may be drawn. 



The very appearance of this scale will show its construction, the 
side of the square a b may be of any length whatever, it is gener- 
ally taken an inch, but this is not imperative. 

By means of the 10 parallel lines running along the length of the 
scale, and the 10 diagonal lines parallel to each other in the square 
a b c d, we have 100 intersections in the square, by which we are 
enabled to find any and every hundredth part of the division of a b. 

For example, I wish to find 27 hundredths of the line a b. I go 
to the division 2 on a b, and then run up that diagonal line to the 7th 
parallel, and from that intersection to the line a d is 27 hundredths 
of a b. 

The distances ab, a g, g h, &c, may be taken to represent 1, 10, 
100, or in fact any number we please. Suppose we take any one 
of the equal divisions ab, ag, &c, to represent 100, and then require 
234. From I to e represents that distance. 

If the base a b were 10, from I to ewould be 23.4 ; if 1, then from 
I to e would be 2.34 ; and so on proportionally for any other change 
of base, or change of the unit. 

To transfer distances from the scale (as I e, p q, &c.) to paper, 
we require 



INTRODUCTION. 



11 



D IVI DEES. 

Dividers are nothing more than a delicate pair of compasses — 
two bars turning on a joint. They are too well known to require 
representation by a figure. 

They are also used for describing circles and parts of circles. 

THE PROTRACTOR. 

The following diagram accurately represents this instrument. It 
consists of a semicircle of brass ABC, divided into degrees. 




The degrees are numbered both ways, from A to B and from B 
to A. There is a small notch in the middle of AB, to indicate the 
center. 

To lay off an angle. Place the diameter AB on the line, so that 
the center shall fall on the angular point. 

Then at the degree required, at the edge of the semicircle make 
a point with a pin. Then remove the protractor and draw a line 
through the point so marked and the angular point ; this line, with 
the given line, will make the required angle. 

The reader will observe a great similarity between this instrument 
and the circumferentor, which is described in a subsequent portion 
of this work. 

This instrument is designed merely to draw angles on paper, that 
to draw lines marking given angles, with other lines, in the field. 



12 



SURVEYING. 



In addition to this, both the protractor and circumferentor may be 
used in taking levels, and measuring angles of altitude, when no 
better instruments for such purposes are at hand. 

For instance, if a delicate plumb should be suspended from the 
center of the protractor, and the thread rest at the point C, while the 
instrument is held in a frame, then A and B would be as a level, 
and as many degrees as the plumb line rested from C so 
many degrees would be the inclination of A and B from a horizontal 
level. 

Levels and angles of altitudes were formerly taken in this way. 

With the instruments previously described, solve the following 
problems. The references are to Eobinson's Geometry. Thus, 
(th. 15, b. 1, cor. 1,) indicates theorem 15, book 1, corollary 1, 
where the demonstrations of the problem referred to will be found. 



PROBLEM 1. 

To bisect a given finite straight 

Let AB be the given line, and from its 
extremities, A and B, with any radius 
greater than the half of AB (Post. 3), de- 
scribe arcs, cutting each other in n and m. 
Join n and m; and C, where it cuts AB f 
will be the middle of the line required. 

Proof, (th. 15, b, 1, cor. 1 ). 



A- V*—. B 



PROBLEM 2. 

To bisect a given angle. 

Let ABO be the given angle. With any 
radius, from the center B, describe the arc 
AC. From A and C, as centers, with a 
radius greater than the half of AC, de- 
scribe arcs, intersecting in n; and join Bn, 
it will bisect the given angle. 

Proof, (th. 19, b. 1). 




INTRODUCTION. 



13 



PROBLEM 3. 

From a given point, in a given line, to draw a perpendicular to that 
line. 

Let AB be the given line, and C 
the given point. Take n and m equal 
distances on opposite sides of 0; and 
from the points m and n, as centers, 
with any radius greater than nC or 
ormO, describe arcs cutting each other 
in S. Join SO, and it will be the per- 
pendicular required. Proof, (th. 15, b. 1, cor. 






)• 



The following is another method, which 
is preferable, when the given point, 0, is at 
or near the end of the line. 

Take any point, 0, which is manifestly 
one side of the perpendicular, and join 00; 
and with 00, as a radius, describe an arc, 
cutting AB in m and 0. Join m 0, and produce it to meet the 
arc, again, in n; mn is then a diameter to the circle. Join On, and 
it will be the perpendicular required. Proof, (th. 9, b. 3). 




PROBLEM 4. 

From a given point loithout a line, to draw a perpendicular to that 



Let AB be the given line, and the 
given point. From C, draw any oblique 
line, as On. Find the middle point of 
On by (problem 1), and from that point, 
as a center, describe a semicircle, having 
On as a diameter. From the point m, 
where this semicircle cuts AB, draw Cm, 
and it will be the perpendicular required. 




14 



SURVEYING 



PROBLEM 5. 

At a given point in a line, to make an angle equal to another given 

angle. 

Let A be the given point in the line AB, 
and DCE the given angle. 

From C as a center, with any radius, 
CE, draw the arc ED. 

From A, as a center, with the radius 
AF—CE, describe an indefinite arc; and 
from F, as a center, with FG as a radius, 
equal to ED, describe an arc, cutting the other arc in G, and join 
A G; GAF will be the angle required. Proof, (th. 5, b. 3). 




PROBLEM 6. 

From a given point, to draw a line parallel to a given line. 

Let A be the given point, and CB the 
given line. Draw AB, making an angle, 
ABC; and from the given point, A, in the 
line AB, draw the angle BAD— ABC, by 
the last problem. 

AD and CB make the same angle with AB; they are, therefore, 
parallel. (Definition of parallel lines). 




PROBLEM 7. 

To divide a given line into any number of equal parts. 
Let AB represent the given line, and 
let it be required to divide it into any 
number of equal parts, say five. From 
one end of the line A, draw AD, inde- 
finite in both length and position. Take 
any convenient distance in the dividers, 
as Aa, and set it off on the line AD; 
thus making the parts Aa, ah, be, &c, equal. Through the last 
point, e, draw EB, and through the points a, b, c, and d, draw 
parallels to eB (problem 6'.); these parallels will divide the line as 
required Proof (th. 17, b. 2). 




INTRODUCTION. 



15 



PROBLEM 8. 

To find a third proportional to two given lines. 

Let AB and A C be any two lines. Place 
them at any angle, and join CB. On the 
greater line, AB, take AD=A C, and through 
D, draw DE parallel to BC; AE is the third 
proportional required. 

Proof, (th. 17, b. 2). 




PROBLEM 9. 

To find a fourth proportional to three given lines. 

Let AB, AC, AD, represent the A 

three given lines. Place the first two 
together, at a point forming any angle, 
as BAC, and join BC. On AB place 
AD, and from the point D, draw 
(problem 6) DE parallel to BC; AE 
will be the fourth proportional required. 

Proof, (th. 17, b. 2). 



B 




PROBLEM 10. 

To find the middle , or mean proportional, between two given lines. 

Place AB and BC in one right line, 
and, on AC, as a diameter, describe a 
semicircle (postulate 3), and from the 
point B, draw BD at right angles to A C 
(problem 3); BD is the mean propor- 
tional required. 

Proof, (scholium to th. 17, b. 3). 




16 



SURVEY 1 MG, 



PROBLEM 11. 

To find the center of a given circle. 

Draw any two chords in the given circle, 
as AB and CD; and from the middle point, 
n, of AB, draw a perpendicular to AB; 
and from the middle point, m, draw a per- 
pendicular to CD; and where these two 
perpendiculars intersect will be the center 
of the circle. Proof, (th. 1, b. 3). 




PROBLEM 12. 

To draw a tangent to a given circle, from a given point, either in 

or without the circumference of the circle. 

When the given point is in the circum- 
ference, as A, draw AC the radius, and 
from the point A, draw AB perpendicular 
to A C; AB is the tangent required. 

Proof, (th. 4, b. 3). 



When A is without the circle, draw 
A C to the center of the circle ; and on 
AC, as a diameter, describe a semi- 
circle ; and from the point B, where 
this semicircle intersects the given 
circle, draw AB, and it will be tangent 
to the circle. 

Proof, (th. 9, b. 3), and (th. 4, b. 3). 




PROBLEM 13. 

On a given line, to describe a segment of a circle, that shall contain 
an angle equal to a given angle. 



INTRODUCTION. 



17 



^^-^p 



Let AB be the given line, and 
the given angle. At the ends of the 
given line, make angles DAB, DBA, 
each equal to the given angle, C. 
Then draw AE, j5i?, perpendiculars to 
AD, BD; and from the center, E, with 
radius, EA or EB, describe a circle ; 
then AFB will be the segment required, as any angle F, made in 
it, will be equal to the given angle, C. 

Proof, (th 11. b. 3), and (th. 8, b. 3). 



PROBLEM 14. 

To cut a segment from any' given circle, that shall contain a given 
angle. 

Let C be the given angle. Take 
any point, as A, in the circumference, 
and from that point draw the tangent 
AB; and from the point A, in the line 
AB, make the angle BAD=C (pro- 
blem 5), and AED is the segment 
required. 

Proof, (th. 11, b. 3), and (th. 8, b. 3). 




PROBLEM 15. 

To construct an equilateral triangle on a given finite straight line. 



Let AB be the given line, and from one 
extremity, i, as a center, with a radius 
equal to AB, describe an arc. At the other 
extremity, B, with the same radius, describe 
another arc. From C, where these two 
arcs intersect, draw CA and CB; ABC y?i\\ 
be the triangle required. 

The construction is a sufficient demonstration. 
2 




Or, (ax. 1). 



18 



SURVEYING. 



PROBLEM 16. 

To construct a triangle, having its three sides equal to three given 
lilies, any two of which shall be greater than the third. 

Let AB, CD, and EF represent the three E F 

lines. Take any one of them, as AB, to be one C D 

side of the triangle. From A, as a center, with 
a radius equal to CD, describe an arc ; and 
from B, as a center, with a radius equal to EF, 
describe another arc, cutting the former in n. 
Join An and Bn, and AnB will be the A 
required. Proof, (ax. 1). 




PROBLEM 17. 

To describe a square on a given line. 

Let AB be the given line, and from the extre- 
mities, A and B, draw A C and BD perpendicular 
to AB. (Problem 3.) 

From A, as a center, with AB as radius, strike 
an arc across the perpendicular at C; and from C, 
draw CD parallel to AB; A CDB is the square 
required. Proof, (th! 21, b.- 1.) 



PROBLEM 18. 

To construct a rectangle, or a parallelogram, whose adjacent sides 
>are equal to two given lines. 

Let AB and AC be the two given lines. A C 

From the extremities of one line, draw per- A B 

pendiculars to that line, as in the last problem ; and from these 
perpendiculars, cut off portions equal to the other line ; and by a 
parallel, complete the figure. 

When the figure is to be a parallelogram, with oblique angles, 
describe the angles by problem 5. Proof, (th. 21, b. 1). 






D 


A 


B 


E 



INTRODUCTION. 19 

PROBLEM 19. 

To describe a rectangle that shall be equal to a given square, and 
have a side equal to a given line. 

Let AB be a side of the given square, and 
CD one side of the required rectangle. 

Find the third proportional, EF, to CD 

and AB (problem 8). Then we shall have, 

CD : AB : : AB : EF 

Construct a rectangle with the two given lines, CD and EF 
(problem 18), and it will be equal to the given square, (th. 13, b. 2). 

PROBLEM 20. 

To construct a square that shall be equal to the difference of two 
given squares. 

Let A represent a side of the greater of two given squares, and 
B a side of the lesser square. 

On A, as a diameter, describe a semi- 
circle, and from one extremity, m, as a cen- 
ter, with a radius equal to B, describe an 
arc, n, and, from the point where it cuts the 
circumference, draw mn and np; np is the 
side of a square, which, when constructed, 
(problem 17), will be equal to the difference 
< of the two given squares. Proof, (th. 9, b. 3, and 36, b. 1.) 

PROBLEM 21. 

To construct a square, that shall be to a given square, as a line, M, 
to a line, N. 

Place M and N in a line, and on the sum describe a semicircle, 
From the point where they join, draw a perpendicular to meet the 
circumference in A. Join Am and 
An, and produce them indefinitely. 
On Am or An, produced, take AB= 
to the side of the given square ; and 
from B, draw BC parallel to mn; 
AC is a, side of the required square. 





20 



SURVEYING 



Besides the numerical scale of equal parts, we have scalps vf 
chords, sines, and tangents, which can be constructed corresponding 
to any radius. 

Such scales of course are not scales of equal parts. 

Such scales are constructed in the following manner. 

Take CA any radius, and describe a semicircle. Draw CD at 
right angles to AB, and draw a tangent line from A. Divide the 
arc AD into equal parts 10, 20, 30, &c, beginning at D, and 
subdivide them as much as required. 

Draw 10 10,-1 
20 20,-30 30, &c, 
all parallel to CD. 

From C to 10 on the | 
line CA, is the sine of 
10°. From C to 20 is I 
the sine of 20° &c. &c. | 

The line 10 10 i« 
the sine of 80°, and CD oi CA is the sine of 90°. 

The distance from A to D is the chord of 90°, from A to 10 is 
the chord of 80°, and from A to 20 is the chord of 70°, and so on 
down. Thus we perceive that we can take off any sine or chord and 
lay it down on a ruler ; and chords and sines thus laid off constitute 
the scale of chords, sines, &c. 

Lines drawn from C through any division of the arc, commencing 
at A to strike the tangent line, will mark off the tangent correspond- 
ing to that arc. Thus, if the angle A CH is 30°, then the line AH 
placed on a scale, will represent the tangent of 30° to the radius CA, 
and thus any other tangent can be laid down on the same scale. 

The scale of chords and sines, as well as the scale of equal parts, 
are to be found on the 



' ; ~ . ■ ' ■ — : — : ^ 




.^^—-~ "SSog 


^0 




/ 


. » 






H 


m 




. s 


30V^ 
/\ 20 \ 






/ 


y^ 


l°1 


1 


's< •... 






B r-c 10 20 3 


o _1 



SECTOR 



The sector is commonly made of ivory, and consists of two arms 
whioh open and turn round a joint at their common extremity. 

For some operations, particularly the projection of solar eclipses, 
the sector is a very useful instrument. 



INTRODUCTION. 21 




The figure before us represents one side of a sector with the plane 
scale only upon it. More than one scale can be put on to a side, 
but we represent but one to avoid confusion. 

The scale must be alike on both arms — and it must commence 
exactly at the joint — hence when near the center the different scales 
crowd each other. 

The two arms of the sector always form two sides of a triangle, 
and by opening and closing them we vary the angle, yet the distance 
across from one arm to the other is always proportional to the sides 
of the triangle. 

The advantage of the sector will appear from the following 
problem. 

A map is before me, its scale is 20 miles to an inch ; I wish to 
find the distance in a right line between two points laid down on it. 

1st. I take one inch in the dividers and open the sector, so that 
the distance between 20 and 20 on the two arms, shall just corres- 
pond to the measure in the dividers, that is, shall be one inch. Let 
the sector lie on the table thus opened. 

2nd. Now take the distance you wish to measure, in the dividers ; 
place one foot on one arm of the sector, and the other foot on the 
other aim; so that the feet of the dividers shall fall on the same 
number on both arms of the sector. 

The number thus marked by the dividers will be the distance 
required. The distance between any other two points may be 
measured on the same map, without any computation whatever. 

For another illustration of the utility of the sector, let us suppose, 
that the sine of 20° is required corresponding to a radius of 6 inches. 



22 SURVEYING. 

Take 6 inches in the dividers, and open the sector so that the sine 
of 90° from arm to arm shall be 6 inches. 

The sector being thus open, take the distance from 20 to 20, on 
the line of sines from arm to arm, in the dividers, and that is the 
distance required. 

aUNTER'S SCALE. 

Gunter's scale is commonly two feet in length, containing the 
plane scale and the scale of sines, chords, and tangents on one side 
of it, and the scale for the logarithms of numbers, sines, and tangents 
on the other. 

This scale is very ingenious, but it is not so much used nor con- 
sidered so important as formerly. 



CHAPTER II 

LOGARITHMS. 



Art 1. Logarithms are exponents. 
Thus, if a 2 =9 

and a 3 = 27 



Then a 5 =243; by multiplying the two 

equations together term by term. 

The exponent 2 of the first equation may be considered the log- 
arithm of 9 ; the exponent 3 the logarithm of 27, and the exponent 
5 the logarithm of the number 243. 

In these equations <z=3 the base of the system. 

By the preceding operation it is obvious that adding the exponents 
2 and 3, corresponds to multiplying the numbers 9 and 27. 

If we take the equation a 5 =243, and divide it by a 2 = 9 member 
by member we shall have 

a 5 ~ 2 =a 3 =27. 

Hence adding exponents (logarithms) corresponds to the multiplica- 



LOGARITHMS. 23 

Hon of their corresponding numbers and subtracting the exponents 
(logarithms) corresponds to the division of their numbers. 

It is this property of logarithms that gives them their utility and 
importance. 

Art. 2i The base of our common system of logarithms is 10, and 
in any equation in the form 10 x =n, x is the logarithm of the num- 
ber n whatever number n may represent. If w=10, then the 
equation becomes 10= 10. Whence x=l because 10 1 = 10. There- 
fore in our common system of logarithms the logarithm of 10 must 
be 1. 

Now because 10° = 1 / 

10 1 = 10 

102 = 100 

103 = 1000 

10 4 = 10000 

&c. &c. ; it is plain that the log- 
arithm of 1 is 0, of 10 is 1, of 100 is 2, &c, every power of 10 
increasing the logarithm by 1 , 

It is also obvious, that every number between 1 and 10 must have 
a fractional or decimal number for its logarithm, and every number 
between 10 and 100 must have one, and some decimal for its 
logarithm. 

In the equation 10*= 3, x is the logarithm of 3, and if we 
multiply this by 10 1 = 10, member by member, we shall have 

10 l +*=30 
Multiply this by 10 1 =10 

and we have 10 2 + x =300. 

These results show that 3, 30, 300, have logarithms containing 
the same decimal number ; x differs from each exponent only 
by whole numbers, and thus, generally ; any number multiplied or 
divided by 10, or any power of 10, will have logarithms containing 
the same decimal part. 

Art. 3* For the general computation of logarithms we refer to 
algebra, and in a work like this we shall only attend to such portions 
of theory as to enable the student to use them understandingly and 
with as much practical facility as possible. 



24 





SURVEYING. 


that the logarithm 


of 


10000 


is 


4,00000 


of 


1000 


is 


3,00000 


of 


100 


is 


2,00000 


of 


10 


is 


1,00000 


of 


t 


is 


0,00000 




i 

1 


is- 


-1,00000 


i 

ro o ' 


= 10-2 


is- 


-2,00000 



For every division of the number by 10 we subtract 1 from its 
logarithm, and when the number comes down to 1, and its logarithm 
of course to 0, if we again divide by 10, making it J^ or 10"~ l , we 
must subtract one from the logarithm, making it — 1. 

The decimal portion of a logarithm is always positive, but the index 
or whole number part of it, becomes minus when the value of the 
number is less than 1. 

Art. it The whole number belonging to any logarithm is called 
its index, a very appropriate term, because it indicates, it points out 
where to place the decimal point between whole numbers and 
decimals. 

The index, or (as some call it) the characteristic, is never put in 
the tables (except from 1 to 100), because we always know what 
it is. It is always one less than the number of digits in the whole 
number. This is obvious from Art. 3. 

Thus, the number 3754 has 3 for the index of its logarithm, 
because the number consists of 4 digits ; that is, the logarithm is 3, 
and some decimal. 

The number 34.785 has 1 for the index of its logarithm, because 
the number is between 34 and 35, and 1 is the index for all num- 
bers between 1 and 100. 

All numbers consisting of the same figures, whether integral, 
fractional, or mixed, have logarithms consisting of the same decimal 
part. (Art. 2.) The logarithms differ only in their indices. 

Thus, the number 7956. has 3.900695 for its log. 

the number 795.6 has 2.900695 " 

the number 79.56 has 1.900695 " 

the number 7.956 has 0.900695 " 



LOGARITHMS. 25 

the number .7956 has — 1.900695 for its log. 
the number .07956 has —2.900695 
&c, <fec. 

For every division by 10, we diminish the index by 1. When 
the index is minus it indicates a decimal number ; but let the learner 
remember that the index only is minus ; the decimal part is always 
positive. 

Art. 5* To take out the logarithm of any number from the 
tables, we only consider the digits ; for the logarithms of 7956, or 
of 7.956, or of .007956, have the same decimal part ; and when 
that decimal part is found we then consider the value of the number 
to prefix the index. 

To prefix the index to a decimal, count the decimal point 1 , and 
each cipher 1, up to the first significant figure, and this is the neg- 
ative index. 

For example, find the logarithm of the decimal .00085. To 
accomplish this we must look for the logarithm of the whole num- 
ber 85, and we find its decimal part to be .929419 ; and now, to 
determine the index, we count one for the decimal point and three 
ciphers, making 4 ; hence, we have Num. .00085 - log. — 4.929419. 

The smaller the decimal, the greater the negative index; and 
when the decimal becomes 0, the logarithm becomes negatively 
infinite. 

Art. 6. The logarithm of any number consisting of four digits 
or less, can be taken out of the table directly and without the least 
difficulty. 

Thus, to find the logarithm of the number 3725, we find the 
number 372 at the side, and over the top we find 5, and opposite 
the former and under the latter we find .571126 for the decimal 
part of the logarithm. The 57, the first two decimal, is under 0, 
which is the same for the whole horizontal column. 

Hence, the logarithm of 3725 is 3.571126 
of 37250 is 4.571126 
of 3.725 is 0.571126 
&c, &c. 

Find the logarithm of 1176. We find 117 at the side, and 6 at 
the top, and opposite the former and under the latter we find .407 



26 SURVEYING. 

The point here demands a cipher, and is put in to arrest attention 
to make the operator look to the next horizontal line below for the 
first two decimals. Thus, we find .070407 for the decimal part 
of the logarithm required. 

Hence, the log. of 1176 is 3.070407 

1. What is the log. of .001176 ? Ans. —3.070407 

2. What is the log. of 13.81 ? Ans. 1.140194 

3. What is the log. of 72.55 ? Ans. 1.860637 

4. What is the log. of .6762 ? Ans. —1.830075 

5. What is the logarithm of the number 834785 ? 

Tins number is so large that we cannot find it in the table, but 
we can find the numbers 8347 and 8348. The logarithms of these 
numbers are the same as the logarithms of the numbers 834700 
and 834800, except the indices. 

834700 log. 5.921530 
834800 log. 5.921582 

Difference, 100 52 

Now, our proposed number, 834785, is between the two preced- 
ing numbers; and, of course, its logarithm lies between the two 
preceding logarithms ; and, without further comment, we may pro- 
portion to it thus, 100 : 85=52 : 44.2 

Or, 1. : .85=52 : 44.2 

To the logarithm 5.921 530 

Add 44 



Hence, the logarithm of 834785 is 5.921574 

the logarithm of 83.4785 is 1.921574 

From this we draw the following rule to find the logarithm of 
any number consisting of more than four places of figures. 

Rule. — Take out the logarithm of the four superior places directly 
from the table, and take the difference behveen this logarithm and the 
next greater logarithm in the table. Multiply this difference by the 
inferior places in the number as a decimal, and add the result to the 
logarithm corresponding to the superior places, the sum will be the 
logarithm required. 

Example. Find the log. of 357.32514. 

The four superior digits are 3573 ; the logarithm of these 



LOGARITHMS. 27 

corresponds to the decimal, .553033, for its decimal pari. The 
inferior digits, taken as a decimal, are 

.2514 

122 



5028 
5028 
2514 
30.6708 
This result shows that 30, or more nearly, 31, should be added 
to the logarithm already found, thus giving .553064 for the deci- 
mal part of the logarithm 357.32514. 

Therefore, as three digits of the given number are whole num- 
bers, the index must be 2, and the logarithm 

of 357.32514 is 2.553064 

of 3573251.4 is 6.553064 

of .035732514 is —2.553064 
The change between the place of the decimal point in a number, 
and the corresponding change of the index to its logarithm, should 
be strongly impressed on the mind of a learner. 

Example 2. What is the log. of 366.25636 ? Ans. 2.563785 

3. What is the log. of 39.37079 ? Ans. 1.595174 

4. What is the log. of 2.37581 ? Ans. 0.375812 

Art. 7» We now give the converse of the last article ; that is, 
we give the decimal part of a logarithm to find its corresponding 
number. 

Taking the decimal in Example 1, (Art. 6,) .553064, we demand 
its corresponding number.* 

The next less logarithm in the table, is .553033, corresponding to 
the figure 3573. The difference between this given logarithm and 
the one next less in the table, is 31 ; and the difference between two 
consecutive logarithms in this part of the table, is 122. Now divide 
31 by 122, and write the quotient after the number 3573. 

* To take out a number from its logarithm, never enter the first part of the 
table between 1 and 100. Go to the main table, as it contains many more 
logarithms. 



28 SURVEYING 



That is, 122)31.(254 

244 



660 
610 



500 
488 



The figures, then, are 3573254, which corresponds to the decimal 
logarithm .553064 ; and the value of these figures will, of course, 
depend on the index to the logarithm. 

If this given logarithm contained an index, such index would 
point out how many of these figures must be taken for whole num- 
bers, the others will be decimals ; thus, if the index had been 4, the 
number would be 35732.54 

If the given decimal had been .553063.67, which is the exact 
converse of example 1, then we should have found that number, 
35732514 ; but we did not give that decimal logarithm, because the 
table contains only six decimal places. From this obvious operation 
we derive the following rule to find the number corresponding to a 
given logarithm. 

Rule. — If the given logarithm is not in the table, find the one next 
less, and take out the four figures corresponding ; and if more than 
four figures are required, take the difference between the given logarithm 
and the next less in the table, and divide that difference by the difference 
of the two consecutive logarithms in the table, the one less, the other 
greater than the given logarithm ; and the figures arising in the quotient, 
as many as may be required, must be annexed to the former figures 
taken from the table. 

EXAMPLES. 

1. Given, the logarithm 3.743210, to find its corresponding num- 
ber true to three places of decimals. Ans. 5536.182 

2. Given, the logarithm 2.633356, to find its corresponding 
number true to two places of decimals. Ans. 429.89 

3. Given, the logarithm — 3.291742, to find its corresponding 
number. Ans. .0019577 



LOGARITHMS. 29 

MULTIPLICATION BY LOGARITHMS. 

Art. 8. If the principle first laid down in (Art. 1) is true, the 
sum of the exponents will be the exponent of the product of any 
number of factors. In other words, 

The sum of the logarithms of any number of factors will be the 
logarithm of the product of those factors. 

N. B. The logarithmic table corresponds to this principle, and we 
may see by the following 

EXAMPLES. 

The log. of 3 (taken from the table,) is 0.477121 
The loff. of 4 " " " " is 0.602060 



Therefore the log. of 1 2 must be 1 .079 1 8 1 

Given, the log. of 7 and the log. of 9, to find the logarithm of 63. 
Because 7X9=63, therefore, 

To log. 7=0.845098 

Add log. 9=0.954243 

Sum 1.799341 

By inspecting the table, we shall find this logarithm stands 
opposite 63, and by this process the logarithms of all the composite 
numbers have been found. In this we may consider that the log- 
arithm pointed out the product 63. 

Hence we have the following rule for obtaining the product of any 
number of factors. 

Rule. — Find the logarithm of each factor, add those logarithms 
together and the sum will be the logarithm of the product. The number 
corresponding to this last logarithm taken from the table, will be the 
product itself. 

EXAMPLES. 



1. To multiply 23.14 by 5.062. 

Numbers. Logs. 

23.14 1.364363 
5.062 0.704322 



Product 117.1347 2.0.68685 



2. To multiply 2.581926 by 
3.457291. 

Numbers. Logs. 

2.581926 0.411944 
3.457291 0.538736 



Product. 8.92648 0.950680 



30 



SURVEYING. 



3. To mult. 3.902 and 597.16 
and .0314728 all together. 



Prod. 



Numbers. 

3.902 
597.16 
.0314728 
73.3333 



Logs. 

0.591287 
2.776091 

-2.497935 



1.865313 



Here the — 2 cancels the 2, and 
the one to carry from the decimals 
is set down. 



4. To mult. 3.586, and 2.1046, 
and 0.8372, and 0.0294 all 
together. 

Numbers. Logs. 

3.586 0.554610 

2.1046 0.323170 

0.8372 —1.922829 

0.0294 —2.468347 



Prod. 0.1057618 —1.268956 



Here the 2 to carry cancels the 
— 2, and there remains the — 1 to 
set down. 



DIVISION BY LOGARITHMS. 

Art. 9, As division is the converse of multiplication we draw the 
following rule for division by use of logarithms. 

N. B. Addition and subtraction is to be understood in the algebraic 
sense. 

Rule. — From the logarithm of the dividend subtract the logarithm 
of the divisor, and the number corresponding to the remainder is the 
quotient required. 



EXAMPLES. 




1. Divide 327.5 by 2207 




log. 327.5 


2.515211 


log. 2207 


3.342028 


Quotient .14839 


—1.173183 


2. Divide .054 by 1.75 




log. .054 


—2.732394 


log. 1.75 


0.243038 


Quotient .030857 


—2.489356 



Art. 10» The preceding examples in multiplication and division 
were adduced only to show the nature of logarithms : had our 
object been results, the common arithmetical operations would have 
been more convenient for some of them ; but there are cases that 
demand the use of logarithms, and such cases mostly occur in 
Involution and Evolution. 



LOGARITHMS. 



31 



Rule for Involution. — Take out the logarithm of the given 
number, and multiply it by the index of the proposed power. Find 
the number corresponding to the product, and it will be the power 
required* 

EXAMPLES. 



1. What is the 2d power of 
351? 

log. 351 2.545307 



Ans. 



123201 



5.090614 



2. What is the cube of 1.72 ? 
log. 1.72 0.235528 
3 

Ans. 5.0884 0.706584 



3. What is the 4th power of 
.0916 ? 

■2.961895 
4 



log. .0916 



Ans. .000070401 —5.847580 

Here 4 times the negative in- 
dex is — 8, adding the 3 to carry 
gives — 5. 



4. What is the 17th power of 
1.04? 

log. 1.04 0.017033 
17 



0.119231 
0.17033 



Ans. 1.9476 0.289561 

N. B. This last example be- 
gins to disclose the utility of log- 
arithms. 



5. What is the 6th power of 
1.037? 

log. 1.037 0.015779 
6 



Ans. 



1.243-f- 0.094674 



6, What is the 21st power of 
2.02? 

2.02 



log. 



Ans. 2584454.6 



0.305351 
21 

.305351 
6.10702 

6.412371 



EVOLUTION 

Art. 11. Evolution is the converse of Involution; hence we 
have the following rule for the extraction of roots : 

Take the logarithm of the given number out of the table. Divide 
the logarithm, thus found, by the index of the required root / then the 
number corresponding is the root sought. l 



32 SURVEYING. 



EXAMPLE S. 



1. What is the cube root of 
125? 

log. 125 3)2.096910 



Ans. 5 0.698970 

3. What is the 4th root of 
751? 

log. 751 4)2.875640 



Ans. 5.235+ 0.718910 



2. What is the cube root of 
200? 

log. 200 3)2.301030 

Ans. 5.848-1- 0.767010 

4. What is the 20th root of 
1.035? 

log. 1.035 20)0.014940 

Ans. 1.001718 0.000747 



5. What is the cube root of the decimal .00048 
log. .00048 —4.681241 

To the inexperienced here would be a difficulty, as the index is 
negative, and the decimal part positive. How then shall we divide 
by 3 ? Add — 2 and +2 to the index ; and this is, in effect, 
adding nothing ; it merely changes the form of the index, thus, 
— 6+2.681241 Now, we can divide by 3, and the quotient is 
—2.893747. The corresponding number, or root, sought, is 
.07829+ Ans. 

Remark. — In the preceding articles we have taught all the preliminary rules 
for the use of logarithms ; "but there is a wisdom beyond rules," and he who 
does not arrive at it, attains only the burdens of knowledge without its bene- 
fits. Rules are necessary through the first rudiments of any science ; but he 
who can instantly fall back on to first principles, and do the most advantageous 
thing at the most advantageous point of time, has a practical tact of the 
highest value. 

Art. 1 2. What follows will refer to no particular rules, but 
will be embraced under general principles, which are far above 
rules. 

To understand logarithms well, it is indispensable to study 
the table, and observe the increase of the logarithms, and com- 
pare that increase with the increase of the numbers. For in- 
stance, the logarithms from 1 to 10, increase by 1, and we must 
go ten times as far, that is, to 100, to obtain another increase of 
1 in the logarithms. 

o 

Hence, logarithms to numbers near to unity, increase very 
fast, and far from unity, increase very slowly ; and this is true 



LOGARITHMS. 33 

whichever side of unity the number may be, above or below. 
For instance, 4 is as much above unity as \ is below unity ; so 
623 is as much above unity as ^ is below unity. 

Now 623 multiplied by ¥ f ^ will produce 1 , therefore the log. 
of 623 added to the log. of ¥ -*- 3 , its reciprocal, must produce the 
log. of 1, or zero. 

Observe the decimal logarithms on page 3, and compare them 
with the logarithms on page 20. Those on page 3 are small in 
value, and vary rapidly; those on page 20 are large in value, and 
vary very slowly. 

Hence it is, practically, more difficult to adjust a log. to its 
number, when the decimal part of a logarithm is small, than 
when it is large ; but Ave can avoid the use of a small decimal 
m a logarithm, as the following artifice will show. 

Let us take Example 4, Art. 11. That is, find the number 
corresponding to the log. 0.000747 

Subtract the log. of 1.01 0.004321 (a) 

The log. of .9918 is —1.996426 (b) 

The two logarithms (a) and (b) added together, will produce 
(0.000747) the given logarithm, and the number corresponding 
to this log. will be the product of the two numbers 1.01, 0.9918, 
or 1.001718. 

Our given log. (0.000747) has a small decimal part, and sub- 
tracting the log. of 1.01, produced a logarithm having a large 
decimal part, and this was our object. The large decimal loga- 
rithm we can carry to the table, and take out the corresponding 
number, to great accuracy, by mere inspection. 

Art. 13. Our table only extends to four figures, three at the 
side, and one at the top of the pages, but by a little artifice, the 
table will serve for any number. 

For instance, suppose that the log. of 101248 was required, 

the uninstructed might infer that it could not be obtained, 

because the table does not extend so far; but by factoring it, as 

follows, we find that it is the product of the two factors, 16 

and 6328. 

3 



34 SURVEYING. 

2)101248 



2)50624 

2)25312 

2)12656 

6328 
Log. of 16 is 1.204120 

Log. of 6328 is 3.801266 



Log. of 101248 therefore is 5.005386 

We may take another artifice (which is the common one) to 
bring this number within the scope of the table. Conceive it 
divided by 1000. Then the quotient would be 101.248, and 
this number is between 101 and 102, and of course within the 
scope of the table, but the first method is the most accurate, 
and therefore the best. 

If we demanded the log. of 101249, a number greater by 
unity than the preceding number, which may be a prime num- 
ber, and not capable of being separated into factors, we can 
obtain the log. of 101248 as before, and then add the quotient 
arising from the following division : 
.4342 9448 

w 

The numerator, or dividend, is the modulus of our system, 
and iVis the number immediately preceding the one whose log. 
is sought. For this example JV== 101248, and 

^i 29 il 8 ==0 .000004. 
101248 

"We carry the division only to the sixth decimal place, as this 
is the limit of the table now under consideration. 

Hence, the logarithm of the number 101249 is 5.005390. 

m, . .43429448 . ,, . , , ,, 

I he expression — is the expression or value of the 

correction corresponding to one of number, but if we wished to 
make a correction 2.3, we would multiply the correction for one 



LOGARITHMS. 35 

(.000004) by 2.3, and this will give .000009 for the correction 
corresponding to an increase of 2.3, thus : 

The log. of 101248 is 5.005386 

Add 2.3 add 9 



The log. of 101250.3 is 5.005395 

The log. of 8091 is 3.908002. What is the log. of 8092 
without using the table? 

8091)0.43429448(0.000054 nearly. 
40455 



29744 




To log. of 8091, which is 


3.908002 


Add 


54 



Log. of 8092 is 3.908056 

If the correction had been for part of a unit, we should have 
added the like part of .000054, and this is the principle, and all 
the principle, of deducing one log. from another. 
Given the log. of 5280 to find that of 52815. 
The last number we shall use as 5281.5, then it will differ 
from the given number by \\. 

5280). 43429448(0.000082 correction for 1. 
42240 41 " i 



11894 .000123 
Log. of 5280 is 3.722634 

Log. of 52815 is 4.722757 

If we factor the number 52815, we shall find that it may be 
regarded as the product of two factors, 3521 and 15. 
Log. of 3521 is 3.546666 

Log. of 15 is 1.176091 



Hence, Log. of 52815 is 4.722757 as before. 

For another example : The ratio between the diameter and 
circumference of a circle is 3.14159265, find the logarithm of 
this number. As the number is less than 10, the index will be 
; but to get the logarithm to proper accuracy, we will con- 



36 SURVEYING. 

ceive the number to be a whole number as far as 31415, and 
then we can call the rest .9265, a decimal. The whole number 
31415, is composed of the two factors, 5 and 6283. Therefore 
To the log. of 6283 3.798167 

Add the log. of 5 .698970 

Log. of 3.1415 is 0.497137 

31415).43429448(0.000014X.92 is 13 nearly. 
31415 



120144 

Whence, the log. of 3.141592 is 0.497150, 

as near as our table of six decimal places can express it. 

N". B. We have formed a table consisting of twelve decimal 
places, (without any indices), which can be used for very- 
delicate cases, and the proper explanations are made in the 
Appendix. 

Art. 14. A person who properly understands the principles 
of logarithms, can find the log. of any number, without a table, 
provided he retains in his memory the value of the modulus 
(.43429448), and the value of the log. of 2, and of 3, and of 7, 
and of 11. 



The log. 


of 2 


is 


0.301030 


The log. 


of 3 


is 


0.477121 


The log. 


of 7 


is 


0.845098 


The log. 


of 11 


is 


1.041393 



The log. of 2 has the same decimal part as 20, 200, or .2 
tenths ; the difference is in the indices. 

The same remark will apply to 3, and all other integers. 

examples. 
1. Find the log. of 112, without the table, the logarithms of 2 
and 7 being given. 

112=4.28=4.4.7=2.2.2.2.7. 
Four times the log. of 2, plus the log. of 7, will be the result. 
Log. of 2 = 0.301030X4=1.204120 
Log. of 7 845098 

Log. of 112 2.049218 Ans, 



LOGARITHMS. 37 

2. Find the log. of 1121, without the use of the table. 

The log. of 1120 is given in the first example, and this cor- 
rected for the increase of unity, will be the result. 

1120).43429448(0.000388. 
To 3.049218 

Add .000388 



And log. of 1121 is 3.049606 Arts. 

3. Find the log. of 99 without the use of the table. 
99=3 2 11 0.477121 



Log. of 9 is 0.954242 

Add log. of 11 1.041393 



Whence log. of 99 is 1.995635 Ans. 

4. There are 5280 feet in a mile. Find the log. of this number 
without the use of the table. 

5280=2 4 3.11.10. 

Whence 4 times the log. of 2 is (Ex. 1) 1.204120 
Log. of 3 0.477121 

Log. of 11. 1C 2.041393 



Whence log. of 5280 is 3.722634 Ans. 

5. The sidereal year consists of 365.25637 mean solar days. 
Find the log. of this number true to 6 -places of decimals, without 
the use of the table. 

N. B. As the number is between 365 and 366, the index will 
be 2. Hence, we shall pay no attention to the index ; the arti- 
fice is to obtain the true decimal part. 

Because 33.11=363=3.11.11. 



33 



SURVEYING. 




Twice the log. of 11 is 


2.082786 


Log. of 3 


0.477121 


oar. of 363 therefore is 


2.559907 



We must now correct this log. for 2 
units, and the decimal .25637, or 2\ 
units nearly. 



Correction for the first unit is 



For the second unit it is 



.43429448 



363 
43429448 



=0.001199 S. 



364 



=0.001193 



And for the fraction ^'i5i 2 -—\ 25637=0.000303 
V 365 / 

Log. of 365.25637 therefore is 2.562602 Ans. 

6. The diameter of the earth is 7912 miles. Find the log, of 
this number without using the table. 

7912=2.2.2.989. But 989 = 990—1. 

The log. of 99 or 990 was found in Ex. 3. Using that result, 
we have 

Log. 990 2.995635 

Correction for 1, subtract 439 

Log. of 989 is 2.995196 

Add 3 times log. 2, or log. 8, .903090 

Log. of 7912 therefore is 3.898286 Ans. 

We think that we have fully illustrated that logarithms can 
be readily found, independently of a table. 

Art. 1 5. The converse, of the last problem is not so obvious. 
When the number is given, the logarithm of that number can 
be found, as we have just shown ; the converse of that problem 
is to find the number corresponding to a given logarithm. We 
must remember the log. of 2, of 3, of 7, and of 11, and also 
know the modulus, but it is better to illustrate by 

EXAMPLES. 

1. Find the number corresponding to the log. 3.470263. 

1st. The log. of 2 is .301030 "j From these we can find 
The log. of 3 is .477121 )■ various other logarithms, 
The log. of 7 is .845098 j but we observe that the de- 



LOGARITHMS. 39 

cimal part of our given log. is very nearly the log. of 3, but the 
index is 3, therefore the number sought must consist of four 
places, the highest figure not quite 3 ; that is, the number sought 
is not quite 3000. It is probably not far from 2900, or not far 
from the double of 1456, which is the double of 729, which is 
the cube of 9. 

Therefore 4.9 3 =2916. But 9=3 2 , or 9 3 =3 6 . 
Twice the log. of 2, is log. 4, 0.602060 

6 times log. of 3 is log. of 729 = 2.862726 

Log. of 2916, therefore is 3.464786 

Which subtract from 3.470263 



0.005477 

If we can find the number corresponding to this last logarithm, 

the product of that number into 2916 will be the number sought. 

We can obtain this, approximately, by means of the modulus. 

mi .43429448 „ , 0.005477 

Thus — =Q, and =n. 

2916 ; Q 

And 2916-f-w= the number sought. 

O 00^477 

In this case 0=0.000149. - — =36.8 nearly. 

0.000149 J 

To 2916 add 36.8, and we have 2952.8 nearly, for the number 

required. But the true number is 2953, the error of about two- 

tenths arises from the imperfection of this last operation. 

To show that there are several methods of solving these 

problems on the same general principle, we will take the same 

problem as before. 

The given log. is 3.470263 

The log. of 30, is 1.477121 



The log. of the unknown factor is 1.993142 
The log. of 100 is 2, hence this logarithm under considera- 
tion must correspond to some number near 98. Now we will 
find the log. of 98 to compare it with this log. 
98=2. 49=2. 7 2 . 
Log. of 98=log. 2+2 log. 7.= 1.991226 

This subtracted from 1.993142 



Ajid we have left 0.001916 



40 SURVEYING. 

Hence, the number sought is 30.98=2940 plus, the correction 
which is found by the following process : 

• 4342944 =0.000148 nearly. ^™=13 nearly. 



2940 " .000148 

Therefore, 2940+13=2953, the true number sought. 

Art. 16. Let the pupil observe that in the last problem the 
given log. (3.470263) was diminished by the log. of 30, and the 
remainder by the log. of 2940, hence, the remaining log. 
(0.001916) is the difference between the given logarithm and 
the logarithm of 2940, and in this sense it is a differential of a 
logarithm, and not a logarithm independent by itself. 

Considered as an independent logarithm, it does not corres- 
pond to 1 3, but to some number a little greater than unity. 

Let us take it as an independent logarithm, and how then 
shall we find the number corresponding ? 

The log. of 1 is 0.00000, and therefore the log. 0.001916 may 
be considered as the difference between itself and the log. of the 
number 1. 

Hence, the correction to the number 1 may be found by the 
same process as we have just found the correction to the num- 
ber 2940. 

The operation is thus : 

^^=.434294. .£™£I?=0.00441S. 



1 0.43429 

Whence the log. of 1.004412 is 0.001916. 

The product of the three factors 30.85.(1.004412) is 2953 
very nearly, as it ought to be. 

From the above operation, we can draw the following rule to 
find the corresponding number to any very small logarithm : 

Rule. — Divide the given logarithm by the modulus, and to the 
quotient and unity. The sum will be the number sought .* 

* We have drawn these rules from mere practical observations, without 
any pretensions to science ; but nevertheless, they conform to strict analyti- 
cal deductions, as found in the differential calculus. 

In that branch of science we shall find the equation dx= ■ in which 

y 

tn represents the modulus of a system of logarithms, dxihe differential value 



LOGARITHMS. 41 

Example. The log. of a certain number is 0.003296, find the 
corresponding number. 

By the rule Q - Q03296 =0.007589. Ans. 1.007589. 

J .43429 

For another example. The log. of some number just above unity 
is 0.000123. What is that number? 

By the rule a00Qt — =0.0002832. Ans. 1.0002832. 

J .43429 

We will give but one more example, independent of the tables, 
and the object of that one is to show the utility of logarithms, 
notwithstanding we may be under the necessity of computing 
the logarithms expressly for the example, which is the following: 

Find the value of the fifth power of 8, multiplied by the third 
root of 7, and that product divided by the fifth root of 6. 

By logarithms it will stand thus : 

5Xlog. 8+1 log. 7—i log. 6=\og.x. 

The log. of 8=3 times the log. of 2, which is .903090. 
5Xlog.8 - - - 4.515450 

ilog.7 - 0.281699 

4.797149 
ilog. 6 - - - 0.155630 

Log. x 4.641519 

We must now find the value of x. The index is 4, therefore 
there must be five places for whole numbers. The decimal part 
of the log. is a little greater than the log. of 4, therefore the 
number sought must be a little greater than 40000. It is not 
44000. For the log. of 11-j- the log. of 4 will give a greater 
decimal than the decimal .641519. 

\£ a logarithm, dy the differential value of the number, and y the number 
tself. 
The last operation in (Art. 15), #=2940, m=.434294. 

"=^^=0.000148. ^=0.001916. dy =l. dx== ^^ 13 nearly. 
y 2940 m .000148 J 

When y=l, then dx=mdy, and dy— — , which is the symbol for the last 

m 

•nle. 



42 SURVEYING. 

The number 42, or 42000, which is probably less than x, has 
three factors, 6, 7, and 1000. 

The log. of 6=log.2-f-log.3, - - .778151 

log. 7, - - .845098 

Log. 1000-f log. 42, - 4.623249 

Log. x, - - 4.641519 

L ° g ' 42000 0.018270 

The number corresponding to this remaining log. is found by 
the last rule. 

01 897 
Thus —=0.04205. Log. of 1.04205 is ,018270. 

.43429 6 

Whence — - — =1.04205, or #=43766.1, the approximate 
42000 rr 

number. 

It is not likely, however, that this is the true number, for the 
log. 0.018270, is too great to have its number determined by 
this method, and if greater accuracy is required, we must obtain 
more exact factors. 

We will therefore resume the work from 

log. x =0.018270, 
& 42000 

and we know that the number corresponding to this log. must 

be near 1 .042 , and this number, or a number very near it, can 

be produced by dividing 100 by 96. Or, in other words, this 

log. must be very near the log. of VV • 

But the log. of 100 is 2.000000 

Log. of 96 is log. of 8-f-log. 2+log.6, 1.982271 



0.017729 



Making use of this factor, we shall have 

Log. 96x -=0.000541. 
4200000 

Now we have a log. sufficiently small to determine its number, 
as follows : 

l 000541 =0.001246. Log. of 1.001246=0.000541. 
.43429 ° 



LOGARITHMS. 43 

Q6r 

Whence ___=1.001246. 

4200000 

Or =100.1246. Or #=43804.516, 

7000 

the true number sought. 

But by the help of the table this problem would have been a 
very trifling affair ; but if a person can work it without the table 
by recollecting a few simple logarithms, we are sure that such a 
person has a clear comprehension of logarithms, and it is to 
make this test that we require any one to work without the table. 

The following examples are given to show the practical value 
of logarithms. The student may now use the table. 

EXAMPLES. 

1. What is the cube root of 12326391? Ans. 231. 

The log. of 12320000 is 7.090611 (See table.) 

Correction for ,6391 225 



Log. of 12326391 3)7.090836 

Log. of 231 2.363612 

2. What is the cube root of 592.704? Ans. 8.4. 

3. What is the cube root of 997002999? Ans. 999. 

4. What is the cube root of 40? Ans. 3.41995-)-. 
The utility of logarithms is more strikingly illustrated by 

examples in powers and roots higher than the third, like the fol- 
lowing : 

5. What is the 5th root of 130691232? Ans. 42. 
Log. of 130691232 is 8.116246(5 

Log. of 42 is 1.623249 

6. What is the 6th root of 12230590464? Ans. 48. 

7. What is the 7th root of 10? Ans. 1.38949. 

8. The 10th root of a certain number is .021, what is that 
number? Ans. 0.0000.0000.0000.00001 6684+. 

9. The 10th root of a certain number is 21, what is that 
number? Ans. 16684000000000, nearly. 



14 SURVEYING 



CHAPTER III. 

ELEMENTARY PRINCIPLES OF PLANE 
TRIGONOMETRY. 

Trigonometry in its literal and restricted sense, has for its object, 
the measure of triangles. When the triangles are on planes, it is 
plane trigonometry, and when the triangles are on, or conceived to 
be portions of a sphere, it is spherical trigonometry. In a more 
enlarged sense, however, this science is the application of the prin- 
ciples of geometry, and numerically connects one part of a magni- 
tude with another, or numerically compares different magnitudes. 

As the sides and angles of triangles are quantities of different 
kinds, they cannot be compared with each other ; but the relation 
may be discovered by means of other complete triangles, to which 
the triangle under investigation can be compared. 

Such other triangles are numerically expressed in Table II, and 
all of them are conceived to have one common point, the center of 
a circle, and as all possible angles can be formed by two straight 
lines drawn from the center of a circle, no angle of a triangle can 
exist whose measure cannot be found in the table of trigonometrical 
lines. 

The measure of an angle is the arc of a circle, intercepted be- 
tween the two lines which form the angle — the center of the arc 
always being at the point where the two lines meet. 

The arc is measured by degrees, minutes, and seconds, there being 
360 degrees to the whole circle, 60 minutes in one degree, and 60 
seconds in one minute. Degrees, minutes, and seconds, are desig- 
nated by °, ', ". Thus 27° 14' 21", is read 27 degrees, 14 min- 
utes, and 21 seconds. 

All circles contain the same number of degrees, but the greater 
the radii the greater is the absolute length of a degree ; the cir- 
cumference of a carriage wheel, the circumference of the earth, or 
the still greater and indefinite circumference of the heavens, have 
the same number of degrees ; yet the same number of degrees in 
each and every circle is precisely the same angle in amount or 
measure. 



PLANE TRIGONOMETRY. 45 

As triangles do not contain circles, we can not measure triangles 
by circular arcs ; we must measure them by other triangles, that is, 
by straight lines, drawn in and about a circle. 

Such straight lines are called trigonometrical lines, and take par- 
ticular names, as described by the following 

DEFINITIONS. 

1. The sine of an angle, or an arc, is a line drawn from one end 
of an arc, perpendicular to a diameter drawn through the other end. 
Thus, BF is the sine of the arc AB, and also of the arc BDE. BK 
is the sine of the arc BD, it is also the cosine of the arc AB, and 
BF, is the cosine of the arc BD. 

N. B. The complement of an arc is what it 
wants of 90° ; the supplement of an arc is 
what it what it wants of 180°. 

2. The cosine of an arc is the perpendicu- 
lar distance from the center of the circle to 
the sine of the arc, or it is the same in mag- 
nitude as the sine of the complement of the 
arc. Thus, CF, is the cosine of the arc AB; but CF—KB, the 
sine of BD. 

3. The tangent of an arc is a line touching the circle in one 
extremity of the arc, continued from thence, to meet a line drawn 
through the center and the other extremity. 

Thus, AH is the tangent to the arc AB, and DL is the tangent 
of the arc DB, or the cotangent of the arc AB. 

N. B. The co, is but a contraction of the word complement. 

4. The secant of an arc, is a line drawn from the center of the 
circle to the extremity of its tangent. Thus, CH is the secant of 
the arc AB, or of its supplement BDE. 

5. The cosecant of an arc, is the secant of the complement. 
Thus, GL, the secant of BD, is the cosecant of AB. 

6. The versed sine of an arc is the difference between the cosine 
and the radius ; that is, AF is the versed sine of the arc AB, and 
DK is the versed sine of the arc BD. 

For the sake of brevity these technical terms are contracted thus : 
for sine AB, we write sin.AB, for cosine AB, we write cos.AB, 
for tangent AB, we write tan.AB, &c. 




46 SURVEYING. 

From the preceding definitions we deduce the following obvious 
consequences : 

1st, That when the arc AB, becomes so small as to call it 
nothing, its sine tangent and yersed sine are also nothing, and its 
secant and cosine are each equal to radius. 

2d, The sine and versed sine of a quadrant are each equal to the 
radius ; its cosine is zero, and its secant and tangent are infinite. 

3d, The chord of an arc is twice the sine of half the arc. Thus, 
the chord BG, is double of the sine BF. 

4th, The sine and cosine of any arc form the two sides of a 
right angled triangle, which has a radius for its hypotenuse. Thus, 
CF, and FB, are the two sides of the right angled triangle CFB. 

Also, the radius and the tangent always form the two sides of a 
right angled triangle which has the secant of the arc for its hypo- 
tenuse. This we observe from the right angled triangle CAH. 

To express these relations analytically, we write 

sin. 2 +cos. 2 =i2 2 (1) 

i2 2 -Ban. 2 =sec. 2 ( 2 ) 

From the two equiangular triangles CFB, CAH, we have 
CF:FB= CA'.AH 

_; , , _. . . JR sin. ,~ % 

That is, cos. : sm=itJ : tan. tan.= (3) 

cos. v ' 

Also, . CF:CB=CA:CH 

That is, . cos : B=E : sec. cos. sec.=i2 2 (4) 

The two equiangular triangles CAH, ODL. give 
CA:AH=DL:DC 

That is, . B : tan.=cot : R tan. cot.=i2 2 (5) 

Also, . CF:FB=DL:DC 

That is, . cos. : sin. = cot: .72 cos. i?=sin..cot. (6) 

By observing (4) and (5), we find that 

cos. sec.=tan. cot. (7) 

Or, . cos. : tan. = cot. : sec. 

The ratios between the various trigonometrical lines are always the 
same for the same arc, whatever be the length of the radius ; ana 
therefore, we may assume radius of any length to suit our conven- 
ience ; and the preceding equations will be more concise, and more 



PLANE TRIGONOMETRY. 47 

readily applied, by making radius equal unity. This supposition 
being made, the preceding become 





sir 


i. 2 -}-cos. 2 = 


= 1 


0) 






l+tan. 2 = 


=sec. 3 


(2) 


sm. 

tan.= — 

cos. 


(3) 




1 

cos.= 

sec. 


(4) 


tan.= — - 
cot. 


(5) 




cos.=sin. cot. 


(6) 



The center of the circle is considered the absolute zero point, and 
the different directions from this point are designated by the different 
signs + and — . On the right of C, toward A, is commonly 
marked plus (+), then the other direction, toward E, is necessarily 
minus ( — ). Above AE is called (+), below that line ( — ). 

If we conceive an arc to commence at A, and increase contin- 
uously around the whole circle in the direction of ABB, then the 
following- table will show the mutations of the signs. 

sin. cos. ten. cot. sec. cosec. vers. 

1st quadrant, -f- + -f" + + + + 

2d" + ___-_ + + 

3d" —_ + +:_ — 4- 

4th __ + __-]-__ _j_ 

PROPOSITION 1. 

The chord of 60° and the tangent 45° are each equal to radius; 
the sine of 30° the versed sine of 60° and the cosine of 60° are each 
equal to half the radius. 

(The first truth is proved in problem 15, book 1). 

On C=, as radius, describe a quadrant; take AD=45°, AB 
=60°, and ^£=90°, then £E=30°. 

Join AB, CB, and draw Bn, perpendicular to CA. Draw Bm, 
parallel to AC. Make the angle CAff=90°, and draw CDH. 

In the A ABC, the angle ACB=Q0° 
by hypothesis ; therefore, the sum of the 
other two angles is ( 1 80—60) = 1 20° . But 
CB— CA, hence the angle CBA= the angle 
CAB, (th. 15 b. 1 ), and as the sum of the two 
is 120°, each one must be 60°; therefore, 
each of the angles of triangle ABC, is 60° 




48 SURVEYING. 

and the sides opposite to equal angles are equal ; that is, AB, the 
chord of 60°, is equal to CA, the radius. 

In the A CAH, the angle CAHis a right angle ; and by hypoth- 
esis, A CH. is half a right angle ; therefore, AHC, is also half a right 
angle ; consequently, AH=AC, the tangent of 45°== the radius. 

By th. 15, book 1, cor. Cn—nA; that is, the cosine and versed 
sine of 60° are each equal to the half of the radius. As Bn and 
EC are perpendicular to A C, they are parallel, and Bm is made 
parallel to On; therefore, Bm= On, or the sine 3Q°, is the half of 
radius. 

PROPOSITION 2. 

Given the sine and cosine of two arcs to find the sine and cosine of 
the sum, and difference of the same arcs expressed by the sines and co- 
sines of the separate arcs. 

Let G be the center of the circle, CD, the 
greater arc which we shall designate by a, 
and DF, a less arc, that we designate by b. 

Then by the definitions of sines and co- 
sines, DO=sm.a; GO—cos.a; FI=sm.b; 
GI=cos.b. We are to find FM, which is 
=sin.(a-f-&); GM=cos.(a-{- b); 
FP=sm.(a—b); GP=cos.(a—b). 
Because IN is parallel to D 0, the two As GD 0, GIN, are 
equiangular and similar. Also, the A FBI, is similar to GIN; 
for the angle FIG, is a right angle ; so is HIN; and, from these 
two equals take away the common angle IIIL, leaving the angle 
FIIT= GIN. The angles at If and N, are right angles ; therefore, 
the A FHI, is equiangular, and similar to the A GIN, and, of 
course, to the A GD 0; and the side HI, is homologous to IN, 
and D 0. 

Again, as FI=IF, and IK, parallel to FM, 

FH=IK, and HI=KE. 
By similar triangles we have 

GD:D0=GI:IN 

mi. * • r> • * t\t t\t s ^ n>a COS. 5 

That is, It : sm.a=cos.6 : IN, or IN= 5 

Also, GD:G0=FI:FB 




PLANE TRIGONOMETRY. 49 

►™ ^ . ■, -nrT- -i-rrr cos. a sin.6 
That is, B : cos.a=sin.5 : FH, or .##= — -^ 

Also, GD\GO=GI\GN 

m -^ , ^v,-r ^v,t cos.a cos.5 

That is, i2 : cos.a=cos.& : #iv, or GN— B 

si 

Also, GD:DO=FI:IJI 

mi ~ . . , ,.„ xrr sin .a sin. 6 

That is, is : sm.a=sm.6 : 1H, or .///= 



M 

By adding the first and second of these equations, we have 

IJSr-\-FII=FM=sm.(a+b) 

m . • / , ,x sin. a cos. 5+ cos. a, sin. 5 

That is, . sm. (a-\-b)— = 

jtii 

By subtracting the second from the first, we have 

. _. sin.a cos.5 — cos.a wsin.6 
sm. (a— b) = 

By subtracting the fourth from the third, we have 

GJV— IHzsz GM— cos. (a-\-b) for the first member. 

__ / . 7 \ cos. a cos. b — sin. a sin. b 
Hence, . cos.(a-\-o)= ~ 

By adding the third and fourth, we have 

GN+IH= m+NP= GF=cos.(a—b) 

__. , • • . cos.a cos. &4-sin.a sm -^ 

Hence, . cos. (a — b)= - 

Jx 

Collecting these four expressions, and considering the radius 
unity, we have 

sin.(a+5) = sin.a cos.5-f-cos.a s i n .5 (7 1 
rjx J sin. (a — 5)=sin.a cos.5 — cos. a sin.5 (8 
* ' J cos.(a-J-£)=cos.a cos.6 — sin. a sin. b (9 

cos. (a— b)=cos.a cos.6-j-sin.a sin.6 (10) 

Formula (A), accomplishes the objects of the proposition, and 
from these equations many useful and important deductions can be 
made. The following, are the most essential : 

By adding (7) to (8), we have (11); subtracting (8) from (7), 
gives (12). Also, (9)-f-(10) gives (13); (9) taken from (10) 
gives (14). 

sin.(a-f-5)-|-sin.(a — 5) = 2sin.a cos. 5 (11) 

sin.(a-j-6) — sin.(a — 5)=2cos.a sin. b (12) 

cos.(a-j-5)-f-cos.(a — 5)=2cos.a cos.b (13) 

cos. (a— -b) — cos.(a+6)=2sin. a sin.5 (14) 



W 



50 SURVEYING. 

If we put a-\-b=A, and a — b—B, then (11) becomes (15), 
(12) becomes (16), 13 becomes (17), and (14) becomes (18). 

/ A+B \ / A— B 



(0) 



sin.^4-sin.^=2sin. ( =i£ \ C os. ( — _ ] (15) 

sin.A — sin.i?=2cos. ( — - — J sin. ( — ^~~) 0*0 

os.-4-f-cos.i?=2cos. ( — - — j cos. ( — — - ) (17) 

\os.B— cos.^=2sin. (——- ) sin. ( — ~ ) (18) 



sin 
If we divide (15) by (16), (observing that — -=tan. and 

COS. 1 

- T —^=cot.= — as we learn by equations (6) and (5) trigonome- 



try), we shall have 



sin..4-f-sm..B 
sin.A — sin. 2? 



Whence, 



. (A+B\ /A-5n (A+B\ 

Sm ' (-g-J C0S - ( ~2~ ) J an ' ( T ) 



COS 



sin. A-\- sin. B 



•m -■( ± ?) <-m 



/A+B\ (A—B\ 

: sin.J. — sin.^=tan. — —- :tan. — -— 



(19) 



V 2 / V 2 / 

or in words. The sum of the sines of any two arcs is to the differ- 
ence of the same sines, as the tangent of the half sum of the same arcs 
is to the tangent of half their difference. 

By operating in the same way with the different equations in for- 
mula ( 0), we find, 



(•») 



sm..4-l-sin.2? , / A-\-B 

ti ~=tan. [ — -— 

cos.^L-f-cos.ii \ 2 

sin.^L+sin.^ , / A — B 



) 

'£ - : . ■ t ( ^1- ) 

cos.B — cos. A ' \ 2 / 

sin~4 — sin. 2? / A — B \ 

cos. A -f- cos. B \ 2 / 

sin.^4 — sin.i? _ / A-\-B \ 

cos.B — cos.^l" ' \ 2 / 

cos.^4-t-cos.^_ \ 2 / 
cos.B — cos.-4' 



tn-i^-r) 



(20) 
(21) 
(22) 
(23) 

(24) 



PLANE TRIGONOMETRY M 

These equations are all true, whatever be the value of the arcs 
designated by A and B; we may therefore, assign any possible 
value to either of them, and if in equations (20), (21) and (24), 
we make B= 0, we shall have, 

sin. A A 1 , nr .\ 

— -=tan.- = . (25) 

1+cos.J. 2 cot.^A v ' 

sm.A A 1 , nnS 

=cot.~=7 t- a (26) 



1 — cos. A 2 tan.-|^l 

1-f-cos.^l eot.-|^l_ 1 



(27) 
1 — cos.^4 tan.-^4 tan. 2 -|.4 v 



If we now turn back to formula (A), and divide equation (7) by 
), and (8) b; 
we shall have, 



sin. 
(9), and (8) by (10), observing at the same time, that — -=tan. 



, .,, sin a cos. 5 -f- cos.a sin. 5 

tan.(a-H) — - — - r-- = 

v ' cos.a cos. o — sin. a sm.o 

, , . sin. a cos.5 — cos.a sin.5 

tan.(a — b)= rT — : .- — 

v ' cos. a cos.o-j-sin.a sin. b 

By dividing the numerators and denominators of the second 

members of these equations by (cos.a cos.5), we find, 

sin. a cos. b .cos. a sin.6 

, , ,. cos.a cos. 6 cos.a cos. 6 tan.a-f-tan.5 

tan.(a-H)= ; = ^=; — r 1 — * (28) 

v cos. a cos.o sm.a sin.o 1 — tan.atan.o v ' 

cos.a cos.6 cos.a cos.S 
sin. a cos.b cos.a sin.6 



; ,, cos.a cosi> cos.a cos. $ tan. a — tan.6 
tan.(a— 5)= - — : r— ttz ; — * (29) 

v 7 cos.a cos.6 sm.a sm.o 1 -f-tan.a tan.o v ' 

cos.a cos.6 cos.a cos.6 
If in equation (1 1 ), formula (B), we make a=b, we shall have, 

sin.2a=2sin.a cos.a (30) 
Making the same hypothesis in equation (13), gives, 

cos.2a-j-l=2cos 2 .a (31) 

The game hypothesis reduces equation (14), to 
1 — cos.2a=2sin 2 .a (32) 

The same hypothesis reduces equation (28), to 
2tan.a , an . 



52 SURVEYING. 

The secants and cosecants of arcs are not given in our table, because 
they are very little used in practice ; and if any particular secant is 
required, it can be determined by subtracting- the cosine from 20 ; and 
the cosecant can be found by subtracting the sine from 20. 



PROPOSITION 3. 

In any right angled plane triangle, we may have the following 
proportions : 

1 st. As the hypotenuse is to either side, so is the radius to the sine 
of the angle opposite to that side. 

2d. As one side is to the other side, so is the radius to the tangent 
of the angle adjacent to the first-mentioned side. 

3d. As one side is to the hypotenuse, so is radius to the secant of 
the angle adjacent to that side. 

Let CAB represent any right 
angled triangle, right angled at A. 
AB and AC are called the sides 
of the A, and CB is called the 
hypotenuse. 

(Here, and in all cases hereafter, we shall represent the angles of a triangle 
by the large letters A, B, C, and the sides opposite to them, by the small letters 
a, b, c.) 

From either acute angle, as C, take any distance, as CD, greater 
or less than CB, and describe the arc DE. This arc measures 
the angle C. From D, draw DF parallel to BA; and from E, 
draw EG, also parallel to BA or DF. 

By the definitions of sines, tangents, and secants, DF is the sine 
of the angle C ; EG is the tangent, CG the secant, and CF the 
cosine. 

Now, by proportional triangles we have, 

CB : BA=CD : DF or, a : c—R : sin. (7^ 

CA : AB=CE : EG or, b : c=B : tan. C }■ Q. E. D. 

CA : CB=CE : CG or, b : a=B : sec.tfj 

Scholium. If the hypotenuse of a triangle is made radius, one 
side is the sine of the angle opposite to it, and the other side is the 
cosine of the same angle. This is obvious from the triangle CDF. 





PLANE TRIGONOMETRY. 53 

PROPOSITION 4. 

In any triangle, the sines of the angles are to one another as the 
sides opposite to them. 

Let ABC be any tri- 
angle. From the points 
A and B, as centers, with 
any radius, describe the 
arcs measuring these an- 
gles, and draw^a, CD, 
and mn, perpendicular to AB. 

Then, . . pa—sm.A, mn=sin.B 

By the similar As, Apa and A CD, we have, 

R : sm.A=b : CD; or, R(CD)=b sin.A (1) 

By the similar As Bmn and BCD, we have, 

R : sm.B=a : CD; or, R(CD)=asm.B (2) 
By equating the second members of equations (1) and (2). 
b sin..4=a sin.Z?. 

Hence, . sin.-4 : sm.B=a : b 

Or, . a : 5=sin A : sin. B) 

Scholium 1. When either angle is 90°, its sine is radius. 

Scholium 2. When CB is less than A (7, and the angle B, acute, 
the triangle is represented by A CB. When the angle B becomes 
B ', it is obtuse, and the triangle is A CB'; but the proportion is 
equally true with either triangle ; for the angle CB'D= CBA, 
and the sine of CB'D is the same as the sine of AB' C. In prac- 
tice we can determine which of these triangles is proposed by the 
side AB, being greater or less than AC; or, by the angle at the 
vertex C, being large as ACB, or small as ACB'. 

In the solitary case in which A C, CB, and the angle A, are given, 
and CB less than A C, we can determine both of the As A CB 
and A CB' ; and then we surely have the right one. 

PROPOSITION 5. 

If from any angle of a triangle, a perpendicular be let fall on the 
opposite side, or base, the tangents of the segments of the angle are to 
one another as the segments of the base. 



I q. e. d. 




54 SURVEYING 

Let ABC be the triangle. Let fall the 
perpendicular CD, on the side AB. 

Take any radius, as Cn, and describe 
the arc which measures the angle C. 
From n, draw qnp parallel to AB. Then 
it is obvious that np is the tangent of the 
angle D CB, and nq is the tangent of the angle A CD. 

Now, by reason of the parallels AB and qp, we have, 
qn : np=AD : DB 

That is, tan. A CD : tan.DCB=AD : DB Q. E. D. 

PROPOSITION 6. 

If a perpendicular he let fall from any angle of a triangle to its op- 
posite side or base, this base is to the sum of the other two sides, as the 
difference of the sides is to the difference of the segments of the base. 
(See figure to proposition 5.) 

Let AB be the base, and from C, as a center, with the shorter 
side as radius, describe the circle, cutting AB in G, AC in F, and 
produce AC to E. 

It is obvious that AE is the sum of the sides A C and CB, and 
AF is their difference. 

Also, AD is one segment of the base made by the perpendicular, 
and BD=DG is the other; therefore, the difference of the seg- 
ments is A G. 

As A is a point without a circle, by theorem 1 8, book 3, we have, 
AEXAF=ABxAG 

Hence, . . AB : AE=AF : AG Q. E. D. 

PROPOSITION 7. 

The sum of any two sides of a triangle, is to their difference, as 
the tangent of the half sum of the angles opposite to these sides, to 
the tangent of half their difference. 

Let ABC be any plane triangle. Then, 
by proposition 4, trigonometry, we have, 

CB : AC=sm. A : sm.B 
Hence, 
CB-\-AC : CB— AC=sm.A-j-sm.B : sm.A—sm.B (th. 9 b. 2) 




PLANE TRIGONOMETRY. 55 

But, tan. ( — - — j : tan. ( — - — J =sin.^4+sin.i? : sm.A — sin.jB 
(eq. (1), trig.) 

Comparing the two latter proportions (th. 6, b. 2), we have, 

CB+AC : CB—AC= tan. ( ^t— ) : tan. ( ^— ) Q. K D. 



PROPOSITION 8. 

Given the three sides of any plane triangle, to find some relation 
which they must hear to the sines and cosines of the respective angles. 

Let ABC be the 
triangle, and let the 
perpendicular fall 
either upon, or 
without the base, 
as shown in the 
figures ; and by 
recurring to theorem 38, book 1, we shall find 

dBjt+^t (1) 

2a K } 

Now, by proposition 3, trigonometry, we have, 
B\ cos. C=b : CD 

Therefore, . CdJ^- (2) 

Equating these two values of CD, and reducing, we have, 

R(a?+V-c>) . . 

cos.G— — - — — — (m) 

2ab v ; 

In this expression we observe that the part of the numerator 
which has the minus sign, is the side opposite to the angle ; and 
that the denominator is twice the rectangle of the sides adjacent 
to the angle. From these observations we at once draw the fol- 
lowing expressions for the cosine A, and cosine B. 

Thus, . cos.A=— ±—±- '- (n) 

2bc v ' 




B _JR(a*+c*-P) 



cos.i?=-^- '- (p) 



56 SURVEYING. 

As these expressions are not convenient for logarithmic compu- 
tation, we modify them as follows : 
If we put 2a=A, in equation (31), we have, 

cos.^4-f- 1=2 cos. 2 \A 
In the preceding expression (n), if we consider radius, unity, 
and add 1 to both members, we shall have, 

tf+c 2 —a 2 



cos. ,4 -J- 1 = 1 



26c 
26c-f6 2 -f-c 2 - 



Therefore, 2 cos. 2 4^4= 

2 26c 

__(5 +c) 2 — a 2 

26c 

Considering (6-f-c ) as one quantity, and observing that we have 

the difference of two squares, therefore 

(6-f c) 2 — a 2 =(6-f-c-f-a)(6-r-c— a); but (6-f-c— a)=5+c+a— 2a 

Hence, . 2 cos. 2 iA J2±±*){^+<^*L \ 

2 26c 

/ 6-f-c-f-a \ i 6-f-c-}-a \ 
Or, . . cos. 2 ^= 

By putting — - — =5, and extracting square root, the final 
result for radius unity, is 



i a l s ( s — a ) 
cos.M-aP^ 

For any other radius we must write, 

' IB 2 s( s —«) 

„ . , , t, IRhis—b) 
By inference, cos.^#=>j 



Also, 



ac 

?e-\I — a 



\Rh(s-c) 



In every triangle, the sum of the three angles must equal 180°; and 
if one of the angles is small, the other two must be comparatively 
large; if two of them are small, the third one must be large. The 
greater angle is always opposite the greater side ; hence, by merely 
inspecting the given sides, any person can decide at once which is the 
greater angle ; and of the three preceding equations, that one should 
be taken which applies to the greater angle, whether that be the par- 
ticular angle required or not ; because the equations bring out the 



PLANE TRIGONOMETRY. 57 

cosines to the angles ; and the cosines, to very small arcs vary so slowly, 
that it may be impossible to decide, with sufficient numerical accuracy 
to what particular arc the cosine belongs. For instance, the cosine 
9.999999, carried to the table, applies to several arcs ; and, of course, 
we should not know which one to take ; but this difficulty does not exist 
when the angle is large ; therefore, compute the largest angle first, 
and then compute the other angles by proposition 4. 

But we can deduce an expression for the sine of any of the angles, 
as well as the cosine. It is done as follows : 

EQUATIONS FOR THE SINES OF THE ANGLES. 
Resuming equation (m), and considering radius, unity, we have, 
a 2 4-6 2 — c 2 

COS. G — ; 

2ab 
Subtracting each member of this equation from 1, gives 

r-, ,a^-(^f) (1) 

Making 2a=C, in equation (32), then a=^0, 

And . . 1— cos.C=2 sin. 2 -|(7 (2) 

Equating the right hand members of (1) and (2), 

. ". 21/Y lab— a 2 — 6 2 +c 2 

2 sm. 2 4(7= — ■ — 

2 2ab 

' 2ab ' 

_(c-\-h — a)(c-\-a — b) 

2ab 

( c ~^b— a \ I c-\-a — 'b \ 

• , „ ' 2" ) ( 2 / 
Or, . . .sm*iC= g. ■ 

But, . C+b - a = c ±^- a and C ±^~l = c +^^ 
2 2 2 2 

Put . — - — —s y as before ; then, 



!(s-a)(s-b) 
ab 
By taking equation (p), and operating in the same manner, we 



S in.JC=7- ( 



Have . . . sin.| J B= N /- ( — ^=^ 
From (.) . . sin.^= N /(ESE3 



58 SURVEYING. 

The preceding results are for radius unity; for anj other 
radius, we must multiply by the number of units in such radius. 
For the radius of the tables, we write R; and if we put it under 
the radical sign, we must write R?; hence, for the sines corres- 
ponding with our logarithmic table, we must write the equations 

thus, . . f^ia^BK 



<*&*£ 



ab 

A large angle should not be determined by these equations, for 
the same reason that a small angle should not be determined from 
an equation expressing the cosine. 

In practice, the equations for cosine are more generally used, 
because more easily applied. 

In the preceding pages we have gone over the whole ground of 
theoretical plane trigonometry, although several particulars might 
have been enlarged upon, and more equations in relation to the 
combinations of the trigonometrical lines, might have been given ; 
but enough has been given to solve every possible case that can arise 
in the practical application of the science. 

By the application of equations (1), (31), and (32), the table 
of natural sines and cosines has been computed. 

The operation is as follows. The sine of 30° is half radius ; 
making the radius unity, equation ( 1 ) gives 
^+cos. 2 30°= 1 : whence cos. 2 30°=f or cos. 30°=^^^ 

From (32) we have, sin-a^/ 1-003 - 2a 

2 
Making 2a=3Q°, then sin. 15°=(-J— ^3)2=0.25881904 

From (31) we have, cos.«=a/ / H^°^ 1 _^ 

Making 2a=30° as before, cos.a=(^-\-^,J^ ) 2 =0.96592582 
Having sine and cosine of 15° the second application of these 

equations will give the sine and cosine of the half of 15°, and so 

on through as many bisections as we please. 




PLANE TRIGONOMETRY. 59 

Being desirous of giving a full exposition of the formation of 
table II, we give the following geometrical demonstration of equa- 
tion 30, by the help of the figure in the margin. 

Let the arc AD=2a 
Then DG=sm. 2a, CG=cos.2a, 
DI = sin. a, AD = 2 sin. a, 
CI =cos. a, DB=2D 0=2 cos. a. 
The angle DBA being at the circum- 
ference, is measured by half the arc AD, 
or by a. 

Now, by applying proposition 4 to the triangle DBG, we have 
sin. DBG : DG=sm. 90° : BD. 

The sin. DBG=sm, a, and sin. 90°= 1, the radius being unity ; 
therefore, the preceding proportion becomes, 

sin. a : sin. 2a=l : 2 cos. a. 
Whence 2 cos. a sin. a=sin. 2a. (Same as eq. 30.) 

PROBLEM. 

Given the sine and cosine of an arc, to find the sine and cosine of 
one half that arc. 

Designate the given arc by 2a, the radius by unity, and whatever 
be the value of a, equation (1) gives 

cos. 2 a-f-sin. 2 a = l ( m ) 

It is proved in proposition 1, that the sine of 30° is half the radius: 
therefore, let 2a=30°, then sin. 2a— 0.5 : and equation 30, just 
demonstrated, gives 

2 cos. a sin. a=0. 5. (n) 

Add (m) and (n), and extract the square root of both members. 

Then cos. a +sin. a = 1.22474486 (0) 

Subtracting (n) from (m), and extracting square root, gives 

cos. a— sin. «=0.70710678 (p) 

By subtracting, and adding (p) and (0), and dividing by 2, we 
find 

sin. a=sin. 15°=0.25881904 
cos. a=cos.l5°=0.96592582 



60 SURVEYING. 

Nowlet2a=15°. Then 

cos. 2 a-}-sin. 2 a=l. 
and 2cos.asin.a=0.25881904 

Operating as before, we find 

sin. a=sin. 7° 30'=0.1305261921 

cos. a=cos. 7° 30'=0.9914447879 

Again, put 2a=7° 30' then as before, 

cos. 2 a-{-sin. 2 a=l 

2 cos. a sin. a=0.1305261921 

These equations give 

sin. a=sin. 3° 45'=0.0654031291 
cos. a=cos. 3° 45'=0.9978589222 
Thus we can bisect the arc as many times as we please. After 
Jive more bisections, we have 

sin. a==sin. 7' 1" 52i'"=0.0020453077 
cos.a=cos.7' 1" 52f '=0.99999799 

As the sines of all arcs under 10', may be considered as coin- 
ciding with the arc, and varying with it, we can now find the sine 
of 1' by proportion. 

Thus, T 1" 521"' :%' :: 0.0020453077 : sin. 1 

Or, 25312.5'" : 3600 : : 

Or, 10125 : 1440 : : 0.0020453077 : sin. 1' 

Whence sin. l'=0.0002908882 

sin.2'=0.0005817764 
sin. 3'=0.0008726646 
In formula (B) equation (11), we find 

sin. (a-\-b) -\-sm. (a — b) =2 sin. a cos. b 
Now, if a=3' and5=l' 

sin. 4'-{-sin. 2'=2 sin. 3' cos. V 

We have already sin. 2' and sin. 3', and cos. Y does not sensibly 
differ from unity, therefore 

sin. 4'=2 sin. 3'— sin. 2'=0.001 1635528 
sin. 5'=2 sin. 4' cos. 1 — sin. 3' &c. &c. to 15' 
When the sine of any arc is known, its cosine can be found by 
the following formula, which is, in substance, equation (1) trigo- 
nometry cos. a => /(l4-sin. a) (1 — -sin. a) 



PLANE TRIGONOMETRY 61 

In formula (A) equation (7) we find that 

sin. (a-{-5)=sin. a cos. 5+cos. a sin. b 
Now, if we make a=30° and 5=4' Then 

sin. a=0.5 cos. a=i % /3=0.8660254 
sin. 5=0.001 16355 cos. 5=0.999999323 
Whence 
sin. (30° 4')=(0.5) (0,999999323) + (0.8660254) (0.00116355) 
=0.499999661 -f- 0.0010007620 

=0.501007281 
Equation (8) gives 

sin. (29° 56')=0.498992041 

When the sine and cosine of any arc are both known, the sine 
and cosine of the half or double of the arc can be determined by 
equation 30; — and thus, from equations (30), (7), (8), (11), and 
(1), the sines and cosines of all arcs can be determined. 

But these sines and cosines are expressed in natural numbers, to 
radius unity, hence they are called natural sines and natural cosines, 
and they are all decimals, except the sine of 90° and the cosine of 
0°, each of which is unity. 

To form table II, we require logarithmic sines, and cosines, which 
are found by taking the logarithms of the natural sines and cosines, 
and increasing the indices by 10, to correspond to the radius of 
10000000000. The radius of this table might have been greater or 
less, but custom has settled on this value. 



To find the logarithmic sine of 1', we proceed thus, 
2 log. — 

To which add 10. 



Nat. sin. 1 =0.0002908882 log. —4. 463 726 



The log. sine of 1', therefore is 6. 463 726 

Nat. sin. 3' =0.0008726646 log. —4. 950 847 

Add 10. 

Log. sin. 3' therefore is 6. 940 847 

Thus the logarithmic sine and cosine of all arcs are found. After 
the logarithmic sine and cosine of any arc have been found, the 
tangent and cotangent of the same arc can be found by equations 
(3) and (5), and the secants by (4); that is, 

R sin. a R cos. a R 2 

tan. a= cot. a= — : sec. a= 

cos a sin. a cos. a 



62 S U R VEYIFG. 

For example, the logarithmic sine of 6° is 9.019235, and its 
cosine 9.997614. From these, find tan., cot., and secant. 





B sin. - 


- 


19.019235 




Cos. .... 


subtract 


9.997614 




Tan. is 


- 


9.021621 




jRcos. - 


- 


19.997614 




Sin. 


subtract 


9.019235 




Cotan. is 


- 


10.978379 




B 2 is - 


- 


20.000000 




Cos. - 


subtract 


9.997674 




Secant is - 


- 


10.002326 


Thus 


we find all the materials for 







TABLE II. 

This table contains logarithmic sines and tangents, and natural 
sines and cosines. We shall confine our explanations to the loga- 
rithmic sines and cosines. 

The sine of every degree and minute of the quadrant is given, 
directly, in the table, commencing at 0° and extending to 45°, at 
the head of the table ; and from 45° to 90°, at the foot of the table, 
increasing backward. 

The same column that is marked sine at the top, is marked cosine 
at the bottom ; and the reason for this is apparent to any one who 
has examined the definitions of sines. 

The difference of two consecutive logarithms is given, correspond- 
ing to ten seconds. Eemoving the decimal point one figure will 
give the difference for one second ; and if we multiply this difference 
by any proposed number of seconds, we shall have a difference 
corresponding to that number of seconds, above the logarithm, 
corresponding to the preceding degree and minute. 

For example, find the sine of 19° 17' 22". 
The sine of 19° 17', taken directly from the table, is 9.518829 
The difference for 10" is 60.2 ; for 1", is 6.02x22 133 

Hence, 19° 17' 22" sine is 9.518952 

From this it will be perceived that there is no difficulty in obtaining 
the sin. or tan., cos. or cot., of any angle greater than 30'. 



PLANE TRIGONOMETRY. 



63 



Conversely. Given the logarithmic sine 9.982412, to find its corres- 
ponding arc. The sine next less in the table, is 9.982404, and gives 
the arc 73° 48'. The difference between this and the given sine, is 8, 
and the difference for l", is .61 ; therefore, the number of seconds cor- 
responding to 8, must be discovered by dividing 8 by the decimal .61, 
which gives 13. Hence, the arc sought is 73° 48' 13". 

These operations are too obvious to require a rule. When the arc 
is very small, such arcs as are sometimes required in astronomy, it is 
necessary to be very accurate ; and for that reason we omitted the 
difference for seconds for all arcs under 30'. Assuming that the sines 
and tangents of arcs under 30' vary in the same proportion as the arcs 
themselves, we can find the sine or tangent of any very small arc to 
great accuracy, as follows : 

The sine of 1', as expressed in the table, is . . 6.463726 
Divide this by 60 ; that is, subtract logarithm . . 1.778151 
The logarithmic sine of 1", therefore, is . . . 4.685575 
Now, for the sine of 17", add the logarithm of 17 . 1.230449 

Logarithmic sine of 17", is 5.916024 

In the same manner we may find the sine of any other small arc. 
For example, find the sine of 14' 21^"; that is, 861"5 

To logarithmic sine of 1", is, 4.685575 

Add logarithm of 861.5 2.93 5255 

Logarithmic sine of 14' 21^" 7.620830 

Without further preliminaries, we may now preceed to practical 




EXAMPLES. 

2. In a right angled triangle, ABC, given 
the base, AB, 1214, and the angle A, 51° 40' 
30", to find the other parts. 

To find BC. 

As radius . . 10.000000 
: tan.A 51° 40' 30" 10.102119 
:: AB 1214 . 3.08 4219 
: BC 1535.8 . 3.186338 
N. B. When the first term of a logarithmic proportion is radius, 
the resulting logarithm is found by adding the second and third loga- 
rithms, rejecting 10 in the index, which is dividing by the first term. 

In all cases we add the second and third logarithms together; which, 
in logarithms, is multiplying these terms together; and from that sum 



64 SURVEYING. 

we subtract the first logarithm, whatever it may be, which is 
dividing by the first term. 

To find A a 

As sin. C, or cos. A 51° 30' 40" - - - 9.792477 

: AB 1214 - - - 3.084219 

: : Radius - - - 10.000000 



: AC1957.7 - - - 3.291742 

To find this resulting logarithm, we subtracted the first logarithm 
from the second, conceiving its index to be 13. 

Let AB C represent any plane triangle, right angled at B. 

1. Given AC 73.26, and the angle ^49° 12' 20"; required the 
other parts. 

Ans. The angle 40° 47' 40", BC55A6, and AB 47.87. 

2. Given AB 469.34, and the angle A 51° 26' 17", to find the 
other parts. 

Ans. The angle C 38° 33' 43", BO 588.5, and A O 752.9. 

3. Given AB 493, and the angle (720° 14' ; required the remain- 
ing parts. Ans. The angle A 69° 46', BC 1338, and AC 1425. 

It is not necessary to give any more examples in right angled 
plane trigonometry, for every distance in the traverse table is but the 
hypotenuse of a right angled triangle, and its corresponding latitude 
and departure form the sides of the triangle. 

If any one should suspect an error in the traverse table, let him test it 
by computing the triangle anew. 

OBLIQUE ANGLED TRIGONOMETRY. 

Of the six parts of a triangle, three sides and three angles, three 
of them must be given and one of the given parts must be a side. 
The subject presents four cases. 

1. When two sides are given, and an angle opposite one of them. 

2. When two sides are given, and the included angle. 

3. When one side and tivo angles are given. 

4. When the three sides are given. 

The principles previously demonstrated are sufficient, and indeed 
ample, to give alb solutions that can come under any one of these 



OBLIQUE ANGLED TRIGONOMETRY. 65 

cases. The operator must use his own judgment in applying these 
principles. 

We give an example in each case, which, with the incidental 
examples, will be sufficient to fix the principles in the mind of the 
operator. 

EXAMPLE 1. 

In any plane triangle, given one side and the two adjacent angles, to 
find the other sides and angle. 

In the triangle ABC, given AB=S76, the angle ^4=48° 3', and 
the angle i?=40° 14', to find the other parts. 

As the sum of the three angles of every triangle is equal to 180°, 
the third angle C must be 180°— 88° 17'=91° 43'. 

INSTRUMENTALITY. 

Take 376 from the scale, by means of the dividers, and place it 
on paper; making one extremity of the line A, and the other 
extremity B. From A, by means of the protractor (or otherwise), 
make the angle -4=48° 3', and from B, make the angle 
i?=40° 14'. The intersections of the lines AC, BG, will give the 
angle C, which being measured will be found to be a little more 
than a right angle. 

Take A C in the dividers, and apply it to the scale, and it will be 
found to be 243 ; and i?Cwill be found to be 279.8, if the projec- 
tion is accurately made ; hut no one should expect numerical accuracy 
from this mechanical method. 

N. B. Our figures in the book do not pretend to accuracy, they should be 
drawn on paper on a larger scale. 

BY LOGARITHMS. 

To find A C. 

As sin. 91° 43' - - - 9.999805 

: AB376 - - - - 2.575188 

: : sin. AB 40° 14' - - - 9.810167 



12.385355 



: AC 243 - - - - 2.385550 

Observe, that the sine of 91° 43' is the same as the cosine of 1 ° 43' 
5 



66 SURVEYING. 

To find BO. 

As sin. 91° 43' - 9.999805 

: AB376 - - - - 2.575188 

: : sin. ^48° 3' - - - 9.871414 



12.446602 




: 5(7279.8 - 2.446797 

EXAMPLE 2, 

In a plane triangle, given two sides, and an angle' opposite one of 
them, to determine the other p arts. 

Let ^4D=1751. feet, one of the given 
sides. The angle Z>=31° 17' 19", and 
the side opposite, 1257.5. From these 
data, we are required to find the other 
side, and the other two angles. 

In this case we do not know whether 
A C or AE represents 1257.5, because 
A C—AE. If we take A C for the other given side, then D is the 
other required side, and DA C is the vertical angle. If we take AE 
for the other given side, then BE is the required side, and DAE is 
the vertical angle ; but in such cases we determine both triangles. 

INSTRUMEN TALLY. 

Draw DE indefinitely — from the point D make the angle 
2)=31° 17'. AD=1751., but. call it 175.1, which take from the 
scale. Place one foot of the dividers at D, the other foot will extend 
to A, thus finding the point A. 

Take 125.75 in the dividers, place one foot at A as a center, and 
with the other strike an arc, cutting DE in and E. Join A 0, 
AE, and one or the other of the triangles A CD ADE, will be the 
triangle required. D and DE applied to the scale, will give one- 
tenth of the required side, and the angle E or D CA, measured, will 
be one of the required angles. 

We can also take one hundredth part of the sides, as well as the 
tenth ; this will make no difference with the angles, the triangles 
thus formed will be similar. 

In that case AD— MM, and the side sought will be 23.64, which 
can be changed to 2364. 



OBLIQUE ANGLED TRIGONOMETRY. 67 

BY L O GARITHM S. 

To find the angle E= C. 
(Prop. 4.) As AC=AE=1257.5 log. 3.099508 

: D 31° 17' 19" sin. 9.715460 

:: AD 17 51 log. 3.243286 

12.958746 



E= C : 46° 18' sin. 9.859238 

From 180° take 46° 18', and the remainder is the angle DC A 
= 133° 42'. 

The angle DAC=ACE—D (th. 1 1, b. 1) ; that is, 
DAC=46° 18'— 31° 17' 19"=15°0' 41". 

The angles D and E, taken from 180°, give DAE=102° 24' 41". 
To find J) C. 
As sin. D 31° 17' 19" log. 9.715460 

: AC 1257.5 log. 3.099508 

: : sin. DAC 15° 0'41" log. 9.413317 

12.512825 



: DC626M 2.797165 

To find DE. 

As sin. D 31° 17' 19 9.715460 

: .4^1257.5 3.099508 

: : sin. 102° 24' 41" 9.989730 



13.089238 



: DE 2364.5 3.373778 

N. B. To make the triangle possible, ^1(7 must not be less than 
AB, the sine of the angle D, when DA is made radius. 

EXAMPLE 3. 

In any plane triangle, given two sides and the included angle, to find 
the other parts. 

Let AD=1751 (see last figure), 7)^=2364.5, and the included 
angle D=41° 17' 19". We are required to find DE, the angle 
DAE, and angle E. Observe that the angle E must be less than 
the angle DAE, because it is opposite a less side. 



68 SURVEYING. 

INSTRUMENTALITY. 

Take DJE=236A5 from the scale (as near as possible), and from 
D draw DA, making the given angle 31° 17' 19". 

Take 175.1 from the scale, in the dividers, and with it mark off 
DA. Join AE ; and ADE will be the triangle in question, and AE 
applied to the scale will give the tenth part of the side sought ; and 
measuring the angle E with the protractor (or otherwise), will 
determine its value. 

BY LOGARITHMS. 

From - - - 180° 

Take D - - - - 31° 17' 19" 



Sum of the other two angles =148° 42' 41" (th. 11, b. 1) 
f sum = 74° 21' 20" 

By proposition 7, 

DE-\-DA : DE—DA=tdin. 74° 21' 20" : tan. \ (DAE—E) 

That is, 

4115.5 : 6l3.5=tan. 74° 21' 20" : ^(DAE—E) 

Tan. 74° 21' 20" - - - 10.552778 

613.5 2.787815 

13.340593 

4115.5 log. (sub.) 3.614423 

\(DAE—E) tan. 28° 1'36" 9.726170 

But the half sum and half difference of any two quantities are 
equal to the greater of the two ; and the half sum, less the half 
difference, is equal the less. 

Therefore, to 74° 21' 20" 
Add 28 1 36 



DAE=102° 22' 5G r 
E= 46 19 44 
To find AE. 
As sin. .#46° 19' 44" - - 9.859323 

: DA 1751 - - - - 3.243286 
: : sin.Z>31°17'19" - - 9.715460 



12.958746 



AE1257.2 - - - 3.099423 



OBLIQUE ANGLED TRIGONOMETRY. 69 

EXAMPLE 4. 

Given the three sides of a plane triangle to find the angles. 
Given .4(7=1751, £5=1257.5, -45=2364.5 
If we take the formula for cosines, we will 
compute the greatest angle, which is (7. 



INSTRUMENTALITY. 

Construct a triangle with the three given 
sides 236.45, 125.75, and 175.1, according 
to problem 16, chapter 1. The angles then measured will show 
their value. 




BY Li 

$=2686.5 

s— c=322 


COS. 


RITHMS . 

log. 
3.099508 
3.243286 


20.000000 
3.429187 
2.507856 


Numerator, 
a 1257.5 
51751. 


25.937043 


Denominator, log. 


6.342794 

2 


6.342694 


i(7= 51° 11' 10" 
(7=102 22 20 


)19. 594249 
9.797124 



The remaining angles may now be found by problem 4. 
We give the following examples for practical exercises : 
Let AB C represent any oblique angled triangle. 

1. Given AS 697, the angle A 81° 30' 10", and the angle 5 40° 
30' 44", to find the other parts. 

Ans. AC 534, 5(7813, and the angle (757° 59' 4". 

2. If ^4(7=720.8, the angle ^4=70° 5' 22", and the angle B= 
59° 35' 36", required the other parts. 

Ans. AS 643.2, 5(7785.8, and the angle (7 50° 19' 6". 

3. Given 5(7980.1, the angle A 7° 6' 26", and the angle 5 106 3 
2' 23", to find the other parts. 

Ans. .45 7284, AC 7613.3, and the angle C 66° 51' 11". 



SURVEYING. 

Surveying is the art of running definite lines on the surface of the 
earth, measuring them, and finding the contents of lands ; and the 
subject necessarily includes the measure of surfaces generally. We 
shall therefore commence with mensuration. 

Mensuration is the application of the principles of Geometry, to 
the measure of surfaces and solids, and when lands are measured it 
is a part of surveying. We shall be very brief on mensuration 
proper, because the rules are so simple and obvious. For the 
demonstration of the rules, we refer to (Legendre and Robinson's 
Geometry.) 

All surfaces are measured by the number of square units which 
they contain. The unit may be taken at pleasure ; it may be an 
inch, foot, yard, rod, mile, &c, as convenience and propriety may 
dictate. 

The square unit is always the square of the linear unit. 

PROBLEM I. 

To find the area of a square, or a parallelogram. 

Rule. — Multiply the length by the perpendicular breadth, and the 
product will be the area. 

(Leg. b. IV, prop. V. Rob. book I, th. 29). 

1. What is the area of a square whose sides are 6 feet 3 inches ? 

Ans. 39^ square feet. * 

2. How many square feet are in a board that is 13| feet long 
and 10 inches wide ? Ans. 11£ square feet. 

3. A lot of land is 80 rods long, and 45 rods wide, how many 
square rods does it contain, and how many acres ? 

Ans. 3600 rods, 22|- acres. 

* Note. — Reductions from one measure to another have no reference to the 
rules here given. 
(70) 



MENSURATION. 71 

4. A man bought a farm 198 rods long, and 150 rods wide, at 
1 32 per acre ; what did it come to ? Ans. $5940. 

PROBLEM II. 

To find the area of a triangle, when the base and altitude are 
given. 

Rule. — Multiply one of these dimensions by half the other, and 
the product will be the area required. 

(Leg. book IV, p. VI. Rob. book I, th. 30). 

1. How many yards in a triangle whose base is 148 feet, and 
perpendicular 45 feet ? Ans. 370 yards. 

2. What is the area of a triangle whose base is 18i feet and alti- 
tude 25i feet ? Ans. 231|i feet. 

PROBLEM III. 

Investigate and give a rule for finding the area of a triangle when 
two sides and their included angle are given. 

Let ABC be the triangles, AB, BO the 
given sides, and B the given angle. 

Represent the side opposite to the angle A, 
by a, opposite C, by c, and opposite B, by b. 

Now a and c are the given sides, and by problem II, the area is 
\a(AD) (1) 

The trigometrical value of AD can be found from the right 
angled triangle ABD. 

Thus, sin. ABB : c : : sin. B : AB. 

That is, 1 : c : : sin. B : AB. 

Whence AB=c sin. B. 

This value of AB substituted in (1) gives 

\ac sin. B= area A (2). 

This expression is the area of the triangle, and from it we draw 
the following rule. 

Rule. — Take half the product of the two given sides and multiply 
it by the natural sine of the included angle, and the last product will be 
the area required. 




72 SURVEYING. 

1. One side of a triangle is 84 feet, another 90 feet, and their 
included angle is 27° 31'. What is the area ? 

Ans. 1746.4 square feet. 
27° 31' Nat. sine. - - - 46201 
\ac - - - - 3780 

3696080 
323407 
138603 



1746.39780 Ans. 
When we use logarithms we have the following rule : 
Rule. — To the logarithms of the two sides, add the log. sine of the 

included angle, and the sum rejecting \0, in the index, is the logarithm 

of twice the area of the triangle. 

2. A certain triangle has one side 125.81, another equal 57.65, 
and their included angle 57° 25', what is its area ? Ans. 3055.7. 
125.81 log. 2.099715 

57.65 log. 1.760799 

57° 25' sine 9.925626 



2 Area, 6111.4 log. 3.786140 

3. How many square yards in a triangle, two sides of which are 
25 and 21 i feet, and their included angle 45° ? Ans. 20.8695. 

PROBLEM IV. 

Investigate and give a rule for finding the area of a triangle when 
the three sides are given. 

(See figure to problem III). Let A represent the area of any 
plane triangle, then by problem III 

A=\ac sin. B. (1) 

But sin. B=2 sin. \B cos. \B. (Eq. 30, trigonometry). 

Therefore, A=ac, sin. \B cos. \B. (2) 

Now in proposition 8, trigonometry, we find 



Sin, 



.fr8= A / («- a >(~> , (3) 



and cos. i J B= A /fi£Z^) (4) 



PLANE TRIGONOMETRY. 73 

The product of (3) into (4) is 



sin. L B cos. ±B=/< s -*)( s - b )( s - c ), (5) 

2 " ^ a 2 c 2 



or ac sin. ^i? cos. ±B=Js(s — a)(s — b)(s — c). (6) 
By comparing (2) and (6) we perceive that 

A=J$(s— a)(s— b)(s— c). 

Here 5 represents the half sum of a, b, and c, therefore, we have 
the following rule to find the area when the three sides are given. 

Rule. — Add the three sides together and take half the sum. From 
the half sum take each side separately, thus obtaining three remainders. 
Multiply the said half sum and the three remainders together; the 
square root of this product is the area required. 

1. Find the area of a triangle whose sides are 20, 30, and 40. 

Ans. 290.47. 

\ sum =45, 1st Rem. =25, 2d=15, 3d=5. 



745.25.15.5= N /225.25.15=15.5 /v /15=75(3.873)=290.474. 

2. How many acres in a triangle whose sides are severally 60, 
50, and 40 rods ? Ans. 6|- nearly. 

3. How many square yards are there in a triangle whose sides 
are 30, 40 and 50 feet ? Ans. 66|. 

4. There is a triangular lot of land containing 8 acres, two of 
its sides are 64, and 46 rods respectively ; what is the angle be- 
tween these sides, and what is the length of the remaining side ? 

Ans, The angle is 60° 25', or is supplement 119° 35'. 
The side is 57.37 rods, or 95.535 rods ; the less angle corres- 
ponding to the lesser side. 

In short, there are two triangles answering 
to the conditions, the one is ABE, the other 
ABC. They are equal because they are on 
the same base and between the same parallels. 
AE = 57.37, A = 95.535. 
7 




74 SURVEYING. 

PROBLEM V. 

To find the area of a trapezoid. 

Rule. — Add the two parallel sides together, and take half the sum. 
Multiply this half sum by the perpendicular distance between the sides. 

Or, The sum of the parallel sides multiplied by their distance 
asunder will give twice the area. 

(Leg. book IV, prop. VII. Rob. b. I, th. 31). 

Remark. — The application of this problem is the most important of any 
in general surveying, as will appear in the sequel, and if the geometrical 
theorem is not familiar to the student he should again review it. 

Ex. 1. In a trapezoid, the parallel sides are 750 and 1225, and 
the perpendicular distance between them 1 540 links : to find the 
area. 

1225 
750 



1975X770=152075 square links =15 acr. 33 perches. 

Ex. 2. How many square feet are contained in a plank, whose 

length is 12 feet six inches, the breadth at the greater end 15 

inches, and at the less end 11 inches ? Ans. 13^f- feet. 

Ex. 3. In measuring along one side AB of a quadrangular field, 

that side, and the two perpendiculars let fall on it from the two 

opposite corners, measured as below, required the content. 

AP = 110 links 

AQ = 745 

AB == 1110 

CP= 352 

DQ= 595 

Ans. 4 acres, 1 rood, 5.792 perches. 

Here we perceive a trapezoid and two right angled triangles. 
N. B. A chain is 4 rods, and contains 100 links ; 10 square 
chains make an acre. 

PROBLEM VI. 

To find the Area, of any Trapezium. 

Divide the trapezium into two triangles by a diagonal ; then find 
the areas of these triangles, and add them together. 




MENSURATION 



75 



Or thus, let fall two perpendiculars on the diagonal from the other 
two opposite angles ; then add these two perpendiculars together, 
and multiply that sum by the diagonal, taking half the product for 
the area of the trapezium. 

Ex. 1. To find the area of the trapezium, whose diagonal is 42, 
and the two perpendiculars on it 16 and 18. 
Here 16+18= 34, its half is 17. 
Then 42 X 17= 714 the area. 

Ex. 2. How many square yards of paving are in the trapezium, 
whose diagonal is 65 feet ; and the two perpendiculars let fall on it 
28 and 33i feet ? Ans. 222 T V yards. 

When the sides of a trapezium, and two of its opposite angles are 
given, the most convenient rule for finding its area is found in 
problem III. 

Conceive CB joined, then the whole figure 
consists of t-wo triangles and the whole area 
is found in the following expression 
(AB X ACX sm.A) + (CD X DB X sin. 
CDB.) 

EXAMPLE. 

In the quadrilateral A CDB we have AC 15.7, CD 20.4, DB 
14.24, and BA 21.1 rods. The angle A 78° 15' and the opposite 
angle CDB 97° 30'. What is the area enclosed ? 

Ans. 356.65 square rods. 

PROBLEM VII. 




To find the area of an irregular figure bounded by any number 
of right lines. 

Rule. — Draw diagonals dividing the figure into triangles. Find 
the areas of the triangles so formed and add them together for the area 
of the whole. 

Let it be required to find the 
area of the adjoining figure of 
five sides. On the supposition 
that AC=36.21 .#(7=39.11 
Aa=4.W Bb=4tmdDd=1.26 
Ans. 296.129. 





76 SURVEYING. 

PROBLEM VIII. 

To find the area of a loner irregular 
figure like the one represented in the 
margin, it is necessary to divide it into 
trapezoids. Then find the area of each 
one of the trapezoids (by problem V.) and add them together for the 
whole area. 

If however the trapezoids have equal distances between their 
parallel sides we can take a more summary proeess, which we dis- 
cover by the following investigation. 

The trapezoid AEFD=\ (a-\-b)XAE. 
EGHF=\(b+c)XEG. 
GIXIT=l(c-\-d)XGI. 
IBCK ^(d+e)XlB. 
On the supposition that AE, EG, GI, &c. are all equal to each 

other the sum of these is / a . , , , e \ 

[^b-{-c+d+-)xAE, 

which represents the area of the whole figure. 

From this we draw the following rule to find the area of a long 

and narrow figure bounded by a right line on one side, and a broken 

or curve line on the other, to which off sets are made at equidistant 

points along the right line. 

Rule. — Add the intemiediate breadths or offsets together, and the half 

sum of the extreme one : then multiply this sum by one of the equal parts 

of the right line, the product will be the area required, very nearly* 
1 . The breadths of an irregular figure at five equidistant places, 

being 6.2, 5.4, 9.2, 3.1, 4.2, and the length of the base 60, what is 

the area ? 

Mean of the Extremes 5.2 

Sum of 5.4, 9.2, 3.1 r7V7 

Sum 22.9 

One of the equal parts 1 5 

1145 
229 
Area=343.5 

* In case DF, FH, &c. are right lines we shall have the area exactly, if they 
are other than right lines the area will be nearly. 



• MENSURATION 77 

2. The length of an irregular figure being 84, and the breadths 
at six equidistant places 17.4 20.6 14.2 16.5 20.1 24.4 ; what 
is the area? Ans. 1550.64. 

PROBLEM IX. 

To find the area of a circle, also any sector or segment of a 
circle. 

Rule 1 . — The area of a circle is found by multiplying the radius 
by half the circumference. 

(Leg. book V, prop. 12. Rob. book V, th. 1.) 

Rule 2. Multiply the square of the diameter by the decimal .7854. 

When the radius of a circle is 1, the length of one degree on the 
circumference is 0.01745 and the whole circumference is 3.1416. 

The radius and the circumference increase and decrease by the 
same ratio, therefore the length of any arc corresponding to any 
radius is easily computed. 

A sector of a circle is to the whole circle as the number of degrees 
it contains is to 360. 

The area of a segment of a circle as FAE, 
may be found by first finding the sector FCE, 
and from it taking the area of the triangle 
FCE. 

This same triangle added to the greater sec- 
tor will give the greater segment. 

These principles and rules are sufficient to 
solve the following examples which are given merely as educational 
Exercises. 

1. What is the area of a circle whose diameter is 10 ? 

Ans. 78.54. 

2. What is the area of a circle whose diameter is 20 ? 

Ans. 4 times 78.54. 

3. What is the area of a circle whose circumference is 12 ? 

Ans. 11.4595. 

4. How many square yards are in a circle whose diameter is 3^ 
feet? Ans. 1.069. 

5. Find the length of an arc of 20°, the radius being 9 feet. 

Ans. 3.141. 




78 SURVEYING. 

6. Find the length of an arc of 60°, the radius being 18 feet. 

Ans. 18.846 

7. To find the length of an arc of 30 degrees, the radius being 9 
feet. Ans. 4.7115. 

8. To find the length of an arc of 12° 10', or 12% the radius 
being 10 feet. Ans. 2.1231. 

9. What is the area of a circular sector whose arc is 18° and the 
diameter 3 feet ? Ans. 0.35343. 

10. To find the area of a sector, whose radius is 10, and arc 20. 

Ans. 100. 

11. Required the area of a sector, whose radius is 25, and its 
arc containing 147° 29'. Ans. 804.3986. 

12. What is the area of the segment, whose height is 18, and 
diameter of the circle 50 ? Ans. 636.375. 

13. Required the area of the segment whose chord is 16, the 
diameter being 20 ? Ans. 44.728. 

14. What is the length of a chord which cuts off one-third of 
the area from a circle whose diameter is 289 ? Ans. 278.6716. 

15. The radius of a certain circle is 10 ; what is the area of a 
segment whose chord is 12? Ans. 16.35. 

16. What is the area of a segment whose height is 2 and chord 
20? Ans. 26.88. 

17. What is the area of a segment whose height is 5, the diame- 
ter of the circle being 8 ? . Ans. 33.0486. 

PROBLEM X. 

To find the Area of an Ellipse. 

Rule. — Multiply the two semi-axes together and their product by 
3.1416. (See conic sections). 

1 . Required the area of an Ellipse whose two semi-axes are 25 
and 20. Ans. 1570.8. 

2. The two semi-axes of an Ellipse are 12 and 9, what is its area ? 

Ans. 339.29. 
To find the area of any portion of a parabola we multiply the 
base by the perpendicular height, and take two-thirds of the product for 
the area required. (See conic sections). 



MENSURATION OF SOLIDS. 79 

Required the area of a parabola, the base being 20, and the alti- 
tude 30. Ans. 400. 

The surfaces of prisms, cylinders, pyramids, cones, &c, are found 
by the application of the preceding rules. 

From theorem 16, book VII, Geometry, we learn that 

The convex surface of a sphere is equal to the product of its diame- 
ter into its circumference. 

The surface of a segment is equal to the circumference of the sphere, 
multiplied into the thickness of the segment. 

In the same sphere, or in equal spheres, the surfaces of different seg- 
ments are to each other as their altitudes. 

MENSURATION OF SOLIDS. 

Br the Mensuration of Solids are determined the spaces included 
by contiguous surfaces ; and the sum of the measures of these 
including surfaces, is the whole surface or superficies of the body. 

The measures of a solid, is called its solidity, capacity, or 
content. 

Solids are measured by cubes, whose sides are inches, or feet, or 
yards, &c. And hence the solidity of a body is said to be so many 
cubic inches, feet, yards, &c, as will fill its capacity or space, or 
another of an equal magnitude. 

The least solid measure is the cubic inch, other cubes being taken 
from it according to the proportion in the following table, which is 
formed by cubing the linear proportions. 

Table of Cubic or Solid Measures. 

1728 cubic inches make 1 cubic foot 

27 cubic feet 1 cubic yard 

166f cubic yards 1 cubic pole 

64000 cubic poles 1 cubic furlong 

512 cubic furlongs 1 cubic mile. 

As the mensuration of solids has little to do with surveying or 
navigation, we shall leave this subject after simply stating the fol- 
lowing truths, which are demonstrated in solid geometry. 

In fact, these truths may be called rules for practical operations. 



80 SURVEYING. 

1. The solidity of a cube, parallelopiped, prism, or cylinder, is 
found by multiplying the area of its base by the altitude. 

2. The solidity of a pyramid or cone is found by multiplying the 
base by the altitude, and taking one-third of the product. 

3. The solidity of the frustum of a pyramid or cone is found by 
calculating the solidity of the pyramid when complete, and subtracting 
from it the solidity of the part removed ;vr find the area of the top 
and bottom of the frustum., and the mean proportional between these 
two areas. Add these three quantities together, and multiply the 
sum by one-third of the altitude of the frustum, and the product 
will be the solidity sought. 

4. Guaging is performed by considering a cash to be made up of 
two frustums of cones placed base to base, and applying the rules for 
the measurement of such solids. 

5. The solidity of a sphere is two-thirds of the solidity of its cir- 
cumscribing cylinder. 



CHAPTER I. 

MENSURATION OF LANDS. 

Lands are not only measured to find their areas, but their exact 
positions must be ascertained, the direction which each line makes 
with the meridians, or with the north and south lines on the earth. 

The boundaries of each tract of land are referred to that meri- 
dian which runs through or by the side of it. 

All meridian lines meet at the poles, therefore they are not 
parallel, (except at the equator,) but the poles are so far distant 
that no sensible error can arise from supposing them parallel, and 
all surveys are made on the supposition that the surface of the earth 
is a plane and the meridians parallel. When large surveys are 
made, like a county or a state, the spherical form of the earth 
should be taken into consideration. 

Meridian lines in surveys are usually determined by the magnetic 
needle, but the needle does not settle exactly north and south, gen- 



MENSURATION OF LANDS. 81 

erally speaking, and the direction which it does settle is called the 
magnetic meridian. 

Surveys are often made by the magnetic meridian as the true one, 
and this would answer every purpose, provided the difference be- 
tween the magnetic and true meridians were every where and at all 
times the same, but this is not so. 

The magnetic meridian is variable, and for this reason it is very 
difficult to trace old lines, unless visible monuments are left, or 
unless the record refers to the true meridian. 

Lines are generally measured by a chain of 66 feet or 4 rods in 
length, containing 100 links, each link is therefore 7.92 inches. 

The area of land is estimated in acres and hundredths, formerly 
in acres, roods, and perches, but the modern method is more simple 
and convenient ; we have a clearer conception of 35 hundreths of 
an acre than we have of 1 rood and 16 perches. 

An acre is equal to 10 square chains or 100,000 square links. 

We may note down the length of a line in chains and hundreths, 
or in links only, for it is nearly one and the same thing : thus, 12 
chains and 38 links may be written 12.38, or 1238 links. 

The area of a field may be found by measuring with the chain 
only, and dividing it into rectangles and triangles, and computing 
each of them separately, according to the rules laid down in men- 
suration. 

The most common method for measuring a field for calculation, 
is, to take the length of all the sides of the field with the chain, and 
their bearings with the surveyor's compass. With these notes an 
accurate plan or plot of the field may be made on paper, and then 
its contents ascertained by cutting it into triangles and measuring 
their bases and perpendiculars with a scale and dividers. A very 
little instruction from a teacher will enable the student to practice 
this method with success ; yet no instrumental measures pretend to be 
numerically accurate, they are but approximately so. 

TO MEASURE A LINE. 

Provide a chain and 10 small arrows or marking pins to fix one 
into the ground, as a mark, at the end of every chain ; two persons 
take hold of the chain, one at each end of it ; and all the 10 arrows 



82 



SURVEYING. 



are taken by one of them who goes foremost, and is called the 
leader ; the other being called the follower, for distinction's sake. 

A picket, or station-staff being set up in the direction of the line 
to be measured, if there do not appear some marks naturally in that 
direction, they measure straight towards it, the leader fixing down 
an arrow at the end of every chain, which the follower always takes 
up, as he comes at it, till all the ten arrows are used. They are 
then all returned to the leader, to use over again. And thus the 
arrows are changed from the one to the other at every 10 chains' 
length, till the whole line is finished ; then the number of changes 
of the arrows shows the number of tens, to which the follower adds 
the arrows he holds in his hand, and the number of links of another 
chain over to the mark or end of the fine. So, if there have been 
3 changes of the arrows, and the follower hold 6 arrows, and the 
end of the line cut off 45 links more, the whole length of the line is 
set down in links thus, 3645. 

In all these measures horizontal distances are required, and they 
are obtained, at least very nearly, by holding the chain in a horizon- 
tal position, both on ascending and descending ground. If the de- 
clivity is too great to admit of measuring a whole chain at a time, 
take a part of it, and in all cases the proper position of the elevated 
extremity should be determined by a plumb line. The reason of 
these operations is obvious by the 
adjoining figure ; we require .the 
fine AB, and not the fine along 
the ground as AC. 

AB=ao-\-cd-\-fC. 

It is not only necessary to 
measure lines but we must also 
know their direction or the angles 
which they make with the meridian. 

This is commonly determined by means of the 
surveyor's compass. 

The surveyor's compass consists of a horizontal circle to which 
are attached sight-vanes and a magnetic needle delicately balanced 
on its centre. 

When the compass is set, that is, standing in a free horizontal 




MENSURATION OF LANDS 



83 



position and the needle free to move on the center, the needle will 
keep the magnetic meridian, and the circular plate may be turned 
under it to bring the sight-vanes to any line ; — the needle will then 
point out the degree of inclination which the line makes with the 
meridian. 

It is important that this part of the subject be most clearly under- 
stood by the learner, we therefore give the following minute illus- 
tration of it. 

Let the reader now face ike north with the booh open before him, his 
right hand is then toward the east, and his left hand toward the 
west. 

The following fig- 
ure represents the 
compass set to the 
magnetic meridian, 
that is the sight-vanes 
Vv and the needle 
lie in the same direc- 
tion. The degrees on 
the plate are num- 
bered both ways from 
.VandSto^and W. 

At the first view 
of this subject, it has 
surprised many to 
find Wior west on the 
right hand toward the 
east, and E in the 
direction toward the west, 
next figure. 

Suppose we wished to find the direction of a line from the center 
of the compass to the object B. We set the compass, that is 
place it horizontal on its staff or tripod, the needle will take the 
same direction as in the first figure, parallel to the margin of the 
paper. 

The sight-vanes Vv are turned toward the object B which turns 
the whole plate, but the needle retains its position. 




The reason of this is explained by the 



84 



SURVEYING. 




We now read the 
degree pointed out 
by the north end of 
the needle, and we 
find it to be about 
50° on the arc 
between N and E, 
showing that the 
course or direction 
from the center of the 
compass to B is North 
about 50° toward the 
East — a result obvi- 
ously true. 

Turning the sight- 
vanes toward the 
north west will bring 
the arc between iYand W to the north point of the needle. For 
example, if it were required to run a line North 31° West, from a 
certain point, all we have to do is to set the compass over that point, 
level the plate, see that the needle is free to move on its pivot, and 
so turn the plate that the north end of the needle will settle at 31° 
between J¥ and W, the range of the sight-vanes will then show the 
required line. Proceed in the. same manner to find any other line. 

Care should be taken that no iron or steel comes near the compass 
while operating with it. To insure a correct position of the needle 
is the principal difficulty, but if it settles with a free motion, descri- 
bing nearly equal arcs, slowly decreasing on each side of a given 
point and finally rests at that point, it operates well, and may be 
relied upon. 

In whatever direction we run, the north point of the needle should 
always lie on some part of the north side of the plate, that is, nearer 
to N than to S and this can always be except when we run due east 
or west per compass. 

Lines should be tested by taking back sights or reverse bearings, 
which will be exactly in the opposite point of the compass, in case 
there is no local attraction to disturb the needle. If the line just 



MENSURATION OF LANDS. 85 

run over does not correspond to the exact opposite point of the com- 
pass, it shows carelessness in running or some local attraction of the 
needle. We shall show how to overcome this last difficulty fur- 
ther on. 

Compasses are usually marked to half degrees, some of them are 
subdivided to one fourth degrees, but by the aid of a vernier scale 
we can theoretically read the arc to one minute of a degree. 

DESCRIPTION OF THE VERNIER. 

The vernier to a compass is on the outer edge of the graduated 
limb. It is a slip of metal made to Jit the graduated limb of an instru- 
ment, and the equal divisions upon it are so made that n divisions on 
the vernier will cover nil divisions on the limb. 

The vernier of the compass is on the outside of the dial plate and 
it is firmly attached to the bar that holds the sight-vanes. The 
dial plate can be moved to and fro along it by means of a screw. 

The vernier is used when the needle points between two divisions 
on the limb : the dial plate is then gently moved by the screw until 
the needle points exactly to the preceding division on the limb, this 
being done, that division on the vernier which makes a right-line, 
that is, coincides with a division on the arc is the number of small 
divisions (minutes) to be added to the division on the limb now 
pointed out by the needle. 

Practical and experienced men, never use the vernier of the compass, 
because they can read the compass without it to greater accuracy 
than they can really run a line. 

But as the vernier scale is of the greatest importance attached to 
several other instruments, which will be referred to in this work, we 
now make an effort to give the learner a clear comprehension of it. 

Let AB represent a 
portion of an arc and ED 
the vernier whicV is con- 
ceived to be attached to 
an index bar and made 
to revolve with it. 

In case on the vernier makes a right line with 10° on the arc 
as is represented in the figure, then the index marks 10°. But if 
on the vernier is a little beyond 10° we then look along the vernier 




86 SURVEYING. 

scale to see what division of it makes a right line with some division 
on the limb, suppose the division 8 on the vernier coincided with a 
division on the limb then the index would mark 10° 8'. 

To understand the philosophy of this : Let x represent the value 
of a division on the vernier and n the number of them which cover 
(n — 1) divisions on the limb, then 

nx=n — 1 

n 

n 

2 

-=2 — 2x 
n 

3 

-=3— Sx 
n 

&c. &c. to n, number, from which it appears that one division on 

the vernier beyond a division on the limb corresponds with the nth 

part of the unit of graduation, two divisions of the vernier above 

two divisions on the limb, correspond with 2nth division of the unit 

of graduation, &c. 

In our figure n=30, the graduation of the limb is to half degrees 

or 30 minutes, and this vernier measures minutes. Verniers on many 

instruments measure as small as ten seconds of arc and on seme 

very large instruments as low as four seconds. 



MENSURATION OF LANDS. 87 



CHAPTER II. 

Having shown in the preceding chapter how to use a compass — 
to run lines, and to measure them ; the next step is to keep a 
proper record of all the lines run, and compute the areas they enclose. 

A line traced on the ground, is called a course, the angle that it 
makes, with the meridian passing through the point of beginning, is 
called its bearing. 

A course written JSf 42° E, indicates that the line runs between 
the north and the east, and makes an angle of 42° with the meridian ; 
when between the north and west, we write N. W., putting the 
number of degrees and minutes between. 

Lines from the south point, are also written S. E. and S. W. ; 
that is, bearings are reckoned from the north and south points, east 
and west, as the case may be. 

Hence, to make a record of a survey, all we have to do is to write 
the bearing and distance of each course, and if the last side runs to 
the point of beginning, it is a complete survey ; otherwise it is not. 

Of course, no area can be attached to any un-enclosed space. To 
complete a partial survey, to enclose a space, to find an area, or to 
test the accuracy of a complete survey ; the most satisfactory method 
of investigation, is that known as 

LATITUDE AND DEPARTURE. 

Latitude is the distance of the end of a line north or south of its 
beginning, measured on a meridian, and it is called either northing 
or southing, according as the line runs north or south. Departure 
is the distance of the end of a line east or west of its beginning, 
measured perpendicular to a meridian, and it is called easting or 
westing, according as the line runs east or west. 

For example, suppose that we have the following bearings and 
distances, which enclose a space represented by AB CD. 

Bearings. Distances. 

AB JV.23°E. - - 17 

BC N.Q3°E. - - 11 

CD S.U°E. - - 23 

DA K 11° W.- - - 23.66 



88 



SURVEYING. 



Let JUS represent the meridian 
running through A, the most western 
point of the field ; make the angle 
JVAB=23°, and AB=17 ; then Ab 
is the latitude, and Bb is the depart- 
ure, corresponding to the course 
AB. By means of the right angled 
triangle ABb, having the hypotenuse 
AB, and the angles, we can com- 
pute Ab, and Bb, 15.65, and 6.64, 
or we can turn to the traverse table, 
and under 23° and opposite 17, we 
shall find the value of these lines at 
once ; and this is the utility of having the traverse table. 

In the same manner we find Bm andm(7, the latitude and depart- 
ure corresponding to the bearing and distance of the line BO. We 
find Bm=1.34, and m(7=10.92. 

Thus we go round the field, taking the latitude and departure of 
each side, and arrange the whole in a table as follows : 




Bearings. 


Dist. 
17 
11 

23 
23,66 


N. | S. 


E. 

6,64 

10,92 

5,56 

23,12 


w. 

23,05 
23,05 


AB 
BO 
OD 
DA 


JST. 23° E- 
K 83° E> 
S. 14° E- 

N. 77° W- 


15,65 
1,34 

5,33 


22,32 




22,32 


22", 32 



When the several operations are performed with perfect accuracy, 
the sum of the northings will be equal to that of the southings, and 
the sum of the eastings to that of the westings. This necessarily 
follows from the circumstance of the surveyor's returning to the 
place from which he set out ; and ifc affords a means of judging of 
the correctness of the work. But it is not to be expected that the 
measurements and calculations in ordinary surveying will strictly 
bear this test. If there is only a small difference, as in the above 
example, between the northings and southings, or between the east- 
ings and westings, it may be imputed to slight imperfections in the 
measurements. 

Here the northings and southings agree, but the eastings are a 
little greater than the westings ; we will therefore decrease the 



MENSURATION OF LANDS. 89 

eastings by half the error, and increase the westings by the same 
amount; the sums will then agree. 

We do this without any formal statement, but the operation is 
strictly that of proportion ; the greater the line the greater the cor- 
rection to be applied. 

When the errors are considerable, a re-survey should be made, 
and if the errors are still great, and in the same direction, there is 
reason to suspect that some local attraction disturbs the free action 
of the needle ; and then, if the importance demands it, a survey 
can be taken without the compass, by methods we shall explain 
in some following chapter. 

We shall make use of this example and this figure to illustrate 
the 

TAKING OF ANGLES BY THE COMPASS. 

For this, and for several other operations in practical mathematics, 
the learner must not expect a written rule : original principles are 
far more simple and reliable. 

We now require the angle ABC; conceive the AB to be pro- 
duced, then the angle between BO and the produced part, is 83° 
less 23°, or 60°. Now 60° taken from 180° gives 120° for the 
angle ABC. 

Again the line BC makes an angle with the meridian toward the 
north of 83°, therefore toward the south it must be 97° on the 
east side of it. The line BA makes an angle of 23° with the meri- 
dian on the west side of it; therefore, the angle ABC — 97+23 
= 120, the same as before. 

To find the angle BCD, we add 83° and 14°. Why ? 

To find the angle CD A, we subtract 14° from 77°. Why ? 

To find the angle DAB, we add 23° to 77°, &c. 

The sum of these 4 angles must equal 4 right angles. 

Suppose now that the surveyor runs the lines AB, B C, CD, and 
then wishes 

TO CLOSE THE SURVEY. 

To close a survey is to run the last side so as to strike the first 
point, when we are not able to see it. 

To accomplish this, we sum up the latitude and departure as far 



90 SURVEYING 

as the point D, the result will show Dd and dA. Having then two 
sides of the right angled triangle, the angle dAD will be the bear- 
ing for dAD = nDA, because nl) and NS are parallel. The side 
DA can also be computed, but it should be measured also, as a test 
to the accuracy of the whole survey. If, on running DA, accord- 
ing to computation, we actually strike the point A, or very near to 
it, and there is little or no difference between actual measure and 
computation, then we may be sure that all the sides have been run 
correctly ; but if on running DA, we do not strike A, or the dis- 
tance does not correspond to computation, we may be sure of errors 
somewhere — either in want of skill or care in the operation, or the 
action of the compass has not been uniform at all the angular points. 
In case of material errors a re-survey should be made. 

In case that we have no means of making computations in the 
field, we may take a course as near the true one as our judgment 
will permit, and run it. This line must bring us near the point of 
beginning, if it does not strike it, and when we get opposite to that 
point we must measure to it at right angles from the fine run ; 
then we shall have data to correct our course. Running a line thus 
by guess work, is called running a random line, from which the true 
line can be found as follows : 

Suppose that when we arrive at D, we judge the course to the 
first point to be N. 75° W., and run that course, and after measuring 
28.65 chains we find that we are passing the first point, which is 83 
finks in perpendicular distance toward the south; what course 
should have been taken ? 

By the following investigation we draw out a rule that dp^Syg 
will apply to all such cases. Hilra 

Let AB represent a true course, AD a random line, [SajHjH 
and DB its amount of deviation. HSU 

Also let DAE equal one degree, and take Ad=l, HHH 
then the deviation at d will be the natural sine of one K|||tf8§ 
degree, and may be taken from the- 'table of natural sines HI 

By proportional triangle we have 

1 :. 01745 \\AD\DE\ 
Whence D 'JS==0. 17 4 5( AD). 

Now DE is contained in DB as often as 1° is contained in the 



MENSURATION OF LANDS. 



91 



number of degrees in the angle DAB. Let x represent the num- 
ber of degrees in DAB, then 

1 M745(AD)' 
That is, x=^fl^ 7 ' 3 \ because .^1^=57.3. 

Hence, to correct a course, we have the following 

Rule. — Multiply the deviation by 57.3, and divide that product by 
the distance, and the quotient will be the number of degrees and parts 
of a degree to add to, or subtract from, the random course. 

This rule, applied to the present example, gives 
. 83(57.3) __ gQ 
23.65 

Hence, the true course is 75°+2°=77°. 

Had the deviation of the random line been toward the south, we 
should have subtracted the correction ; but for this the operator 
must rely on his judgment. 

We now come to the 



COMPUTATION OF AREAS 

By inspecting the figure we per- 
ceive that cCDd is a trapezoid, 
from which, if we subtract the trian- 
gles ADd, ABb, and the trapezoid 
bBCc, the area of the field ABOD 
will be left. 

Observation. — To preserve uniformity 
of expression, and clearness and brevity 
in forming a rule, we shall call triangles, 
trapezoids, while discussing this subject. 
A trapezoid becomes a triangle, when its 
smallest parallel side is so small as to call 
it zero — conversely, then, a triangle is a 
trapezoid, whose smallest parallel side is zero. 

We observe that C is the most northern point of the field, and D 
is the most southern. In traversing from C to D, from the north to 
the south, we pass along the oblique sides of trapezoids that we 
shall call south areas, and in traversing from D to A, B, and C, 




92 SURVEYING. 

from the south to the north, we pass along the oblique sides of 
trapezoids, which we shall call north areas. Now it is obvious that 
if we subtract the sum of the north areas from the sum of the south 
areas, we shall have a remainder equal to the area of the field. 

We now require a systematic method of finding these areas, or 
the area of these several trapezoids. In the first place, we must 
have latitudes and meridian distances. 

Latitude and departure have already been defined and explained. 

MERIDIAN DISTANCES. 

Meridian distances are the distances of the angular points of the 
field from the meridian which runs through the most westerly point 
of the field ; thus, bB, cC, dD, are meridian distances. 

Double meridian distances are the bases of triangles, or the sum of 
the parallel sides of the trapezoids. 

Thus bB is the double meridian distance of the side AB, or it is 
the double meridian distance of the middle point of the line AB. 
bB -f- c C is the double meridian distance of the line B C, or double 
the meridian distance of the middle point of BC. This double 
meridian distance (bB-\-cC) multiplied by bC, or Bm, the latitude 
corresponding to BC, will give the double area of the trapezoid 
bB, Cc. 

We are now prepared to give the following summary or rule for 
finding the area of any field bounded by any number of right lines : 

Rule. — 1 . Prepare a table headed as in the example, namely : 
Bearings, Distance, North, South, East, West, Meridian distance, 
Double meridian distance, North areas, South areas. 

2. Begin at the most western point of the f eld, and conceive a meri- 
dian to pass through that point. 

Find, by the traverse table or by trigonometry, the northings, southings, 
eastings, and westings of the several sides of the field, and set them in 
the table opposite their respective stations, under their proper letters N., 
S., E., or W. 

3. For the first meridian distance take the departure of the first line ; 
for the second, take the first meridian distance and add to it the 
departure of the second line, if the departure is east, or subtract if 
west, &c. 



MENSURATION OF LANDS. 



93 



4. Add each two adjacent meridian distances, and set their sum 
opposite the last of the two in the column of double meridian distances. 

5. Multiply each double meridian distance by the latitude to which it 
is opposite, and set the product in the column of N. areas, if the latitude 
is north, and in that of S. areas, if the latitude is south. 

6. Subtract the sum of the JV. areas from that of the S. areas, and 
take half the remainder, which will be the area of the field in square 
chains. Dividing this by 10 gives the acres ; and the roods and 
rods are found by multiplying the decimal parts by 4 and by 40. 



AB 
BC 
CD 
DA 


Bearings. 

N23 Q E 

N 83° E 
S 14° E 

N77°W 


Dis. 
17 

11 

23 

23.66 


N. 
15.65 

1.34 
5.33 


s. 
22.32 


E. 


w. 

23.05 

(23.08) 


M. D. 

6.63 
17.53 
23.08 

0.00 


D. M.D. 

6.63 
24.16 
40.61 

23.08 


N. areas. 


S. areas 


6.64 

(6.63) 

10.92 

(10.90) 

5,56 
(5.55) 


103.76 
32.37 

123.02 


906.42 












23.08 




259.15 


906.42 


DifF., 

Half, 

Dividing by 10, 


259.15 


647.27 


323.63 
32.363 



Hence the field contains 323 square chains and 63 hundredths, or 
thirty two acres and a little more than 36 hundredths of an acre. 

The numbers in parentheses, as (6.63), and all others in parenthe- 
ses are the numbers corrected to make the eastings and westings 
agree, — the numbers above them are taken from the table. 

Before we give any more examples, it is proper to give some 
examples to show the practical utility of the 

TRAVERSE TABLE. 

This table is computed to every half degree, but if a course is 
between two courses in the table, the operator can use his judgment 
and take out the proper intermediate numbers. 

Those who are not satisfied with this method, can use the table 
of natural sines and cosines, as we shall subsequently explain. 

The distances are consecutive to 30, then 35, 40, &c, to 100. 
But a little thought in the operator will enable him to use the table 
for any distance whatever. 



94 SURVEYING. 

The following examples will illustrate. 

I. A course is JSf. 22° 30' E. distance 62.43 ; what is the corres- 
ponding latitude and departure, as found in the table ? 

"We shall regard the distance as 6243 links, and separate it into 
parts. 

Thus 





Lat. 


Dep. 


6000 


5543 


2296 


240 


221.7 


91.8 


3 


2.77 


1.15 



6243 5767.47 2388.95 

If we now return to chains and links, the latitude is 57.67 and the 
departure 23.89. 

We entered 240 in the table, as 24 chains, and took the numbers 
corresponding. 

2. A course is N. 48° 30' W., distance 187.61 ; what is the corres- 
ponding latitude and departure ? 

Dis. Lat. Dep. 

180.00 119.30 134.80 
7.60 5.036 5.692 
1 066 075 

187.61 124.3426 139.4995 

If we now take the distance as 187 chains and 61 links, the Lat. 
is 124 ch. 34 links, and the Dep. is 139 ch. and 50 links. 

3. A course is S. 81° W., distance 76.87 ; what is the corres- 
ponding latitude and departure ? 

Die. Lat. Dep. 

75.00 1173. 7408. 

1.80 28.2 177.8 
7 1.10 6.91 

76.87 1202.3 7592.71 

If 76 ch. 87 lin., Lat. 12 ch. 2 lin., Dep. 75 ch. 93 links. 
Thus we can find the latitude and departure for any distance cor- 
responding to any degree and half degree. 

We can find it to any degree and minute of a degree by the table 
of natural sines and cosines. 

The common tables containing natural cosines and sines are nothing 
more than latitude and departure corresponding to unity of distance. 



MENSURATION OF LANDS. 95 

Therefore, a double distance will correspond to a double distance 
in latitude and departure, a treble distance will give a treble amount 
of latitude and departure, and so on in proportion. Lat. *= Nat. 
cosine. Dep. = Nat. sine. 

Hence : The natural cosine of any course taken as a decimal, mul- 
tiplied by any distance t will give the latitude corresponding to that 
course and distance. Also, the natural sine taken as a decimal, multi- 
plied by a given distance t will give the departure corresponding to that 
course and distance. 

N. B. Nat. sines and cosines are found in table II., pages 21-65 
of tables. For common purposes, four places of decimals are suffi- 
cient. 

1. The bearing of a certain line is K 35° 18' E. ; distance 12 
chains ; what is the corresponding latitude and departure ? 

Angle 35° 18' N. cos. .81614 
Dis. (multiplier) 12 

Diff. Lat= 9.79368 

2. A certain line runs S. 4° 50' E. ; 
corresponding latitude and departure ? 

Angle 4° 50' N. cos. .9964 
Distance 74.4 

39856 
39856 
69748 



N. sin. 


.57786 
12 




Dep. 


6.93432 




istance 74.40 ; what is 


the 


N. sin. 


.0842 
74.4 

~3368 
3368 
5894 





Lat. 74.13216 Dep. 6.26448 

3. A line makes an angle with the meridian of 75° 41', at a dis- 
tance of 89.75 chains ; what is the latitude and departure ? 

75° 47' cos. .2456 sin. .9694 

Distance 89.75 89.75 



Prod. Diff. Lat. 22.042 Dep. 87.001 

4. A line bearing K 7° 40' W. ; distance 31.20 chains ; required 
the difference of latitude and departure. 

7° 40' cos. .98106 sin. .13341 

Multiplier 31.2 31.2 

Diff. Lat. 30.92 Dep. Tl6~~ 



96 



SURVEYING. 



5. A line running S. 80° 10' E. distance 35.25 chains ; what is 
the difference of latitude and departure ? 

80° 10' cos. .17078 sin. .9853 

Multiplier 35.25 35.25 

Diff. Lat. 6.02 Dep. 34.72 

In the last three examples, we have given only the results of the 

multiplications to two places of decimals ; that is, to the nearest 

link, which is a degree of accuracy sufficient for all practical 

purposes. 

We are now prepared to estimate the areas of the following general 
surveys, given as 

EXAMPLES. 

1. In May, 1845, the following measures of a field were taken. 
Beginning at the western-most point of the field ; thence N. 20° 30' 
E. 5 chains 83 links; thence S. 79° 45' E. 10 chains 15 links; 
thence S. 27° 30' W. 9 chains 45 links ; thence K 63° 15' W. 
8 chains 28 links ; thence JV. 15° 30' W. 1 chain and 4 links, to the 
place of beginning ; required the area. 



It is not absolutely necessary 
to make a plot or figure of the 
field, but for the sake of per- 
spicuity, it is best to do so ; 
yet no reliance is placed on the 
accuracy of the constructed 
figure. 

We perceive by the figure, 
that there are two south areas, 
bBCc, and cCDd ; and three 
north areas, eEdD, AeE, and 
ABb. 

Let the reader observe that 
all the north areas are on the 
outside of the field. 




ENS U RATION OF LANDS 



AB 
BC 
CD 
DE 
EA 


Bearings. 


Dis. 


N. 


s. 


E. 


w. 


M. D. 


D.M.D. 


H . areas. 


8. areas. | 

25.4667 
165.0360' 


IV.20° 30'.E. 
S.79°45'.E. 
S.27°30'W. 
JV.63015'W. 
iV.15°30'W. 


5.83 

10.15 

9.45 

8.28 
1.04 


5.46 

3.73 

1.00 


1.81 
8.38 


2.04 
9.99 


4.36 
7.39 
0.28 


2.04 
12.03 
7.67 
0.28 
0.00 


2.04 

14.07 

19.70 

7.95 

0.28 


11.1384 

29.6535 

0.2800 

41.0719 






10.19 10.19 


12.03 


12.03 




190.5527, 
41.0719 



Area in acres, 



2)1 49.4808 
10)74.7404 

7\472~ 



The operator can rely on the 
rule used in the last operation, 
whatever be the number of sides, 
or whatever be the shape of the 
figure, provided that the lines are 
right lines, from one angular point 
to another. 

In case of re-entering angles, 
like HIA, represented in the ad- 
joining figure, a portion of the 
figure Air, is reckoned twice. But 
this is corrected by the subtractive 
space HIih, which includes not 
only the exterior portion hHIA> 
but also the whole additive trian- 
gle All, belonging to the last side 
of the figure I A. 

In the following example are several such re-entering angles. 

2. Find the area of a lot of land, of which the following are the 
field notes. 

Beginning at the south west corner, the ancient land mark ; thence 
1. K 27° 15' E. distance 9.42 chains. 




2. 


S. 80° 00' E. 


(< 


1.15 


3. 


S. 69° 00' E. 


i. 


12.73 


4. 


S. 15° 45' W. 


tt 


5.00 


5. 


N. 66° 45' W. 


n 


1.05 


6. 


S. 31° 00' IF. 


ti 


2.90 


7. 


iV. 70° 45' IF. 


n 


8.92 


8. 


S. 41° 45' W. 


tt 


2.08 


9. 


N. 63° 00' IK 


tt 


4.18 



SURVEYING 



We can contract the operation in reference to the space it will 
occupy, by putting the difference of latitude in one column, and all 
the departures in another column. 

Marking all the northings by the sign -f*> and all the southings by 
the sign — . Also, all the eastings by the sign -J- , and all the 
westings by the sign — . 

This being understood, the work will appear as follows : 





Bearings. 


Dis. 
9.42 


Lat. 


Dep. | M. D. 


D.M.D. 


N. areas 


S. areas 


1 


N. 27° 15' E. 


-1-8.37 


+. 4.31 4.31 


36.0747 




Q 


1 S. 80° 00' E. 


1.15 


—0.20 


4-1.131 5.44 


9.75 




1.9500 


[- 


\ S. 69° 00' E. 


12.73 


—4.57 


4-11.89,17.33 


22.77 




104.0589 


I 


IS. 15c 45' W. 


5.00 


—4.81 


— 1.36 15.97 


33.30 




160.1730 


i 


) N. 66° 45' W. 


1.05 


4-0.41 


— 0.96J15.01 


30.98 


12.7018 




( 


3 S. 31o 00' W. 


2.90 


—2.49 


pk 1.49 


13.52 


28.53 




71.0397 


" 


1 JV. 70° 45' W. 


8.92 


4-2.94 


— 8.42 


5.10 


18.62 


54.7428 




I 


B S. 41° 45' W. 


2.08 


—1.55 


— 1.38 


3.72 


8.82 




13.6710 


( 


} N. 63° 00' W. 


4.18 
Sum 


4-1.90 


— 3.72 


0.00 


3.72 


7.0680 
110.5873 








0.00 


0.00 




350.8926 


2 
10 


110.5873 


)240.3053 


)120.1526 






12 


acres, a 


nd a small fra( 


jtion over. 


12.01526 



3. Having the following field notes, it is required to find the 



closing side and the area of the field. 



Bearings. 


Dis. 

13.70 
10.30 
16.20 
35.30 
16.00 
9.00 


. N. 


S. 1 


E. 


w. 


1 S. 75° W. 

2 S. 20° 30' W. 

3 W. 
4iV.33° 2tf E. 
5iV.76 v E. 
6 South 


29.51 
3.87 

33.38 
22.19 
11.19 


3.54 
9.65 

9.00 


19.49 
15.52 

35.01 
33.04 

1.97 


13.24 

3.60 

16.20 






22.19 


33.04 



This result shows, that if we commence at the first station, and 
traverse round to the sixth, we shall then be 11.19 chains to the 
north of the place of beginning, and 1.97 chains east of it. This is 
sufficient data to compute the course and distance. 

To compute the area, however, it is not necessary to find either the 
course or the distance. 

We do know, however, by merely inspecting the traverse table, 



MENSURATION OF LANDS. 



99 



that the course to the place of beginning, is south about 10° 20' west, 
and distance near 12 chains. 

We are now prepared to compute the area, and as we wish to 
commence at the western-most point of the field, we shall begin at 
the 4th station, calling it the first : thus, 



Bearings. 


Dis. 
35.30 
16.00 

9.00 

13.70 
10.30 
16.20 


Lat. 
4-29.51 
-4- 3.87 

— 9.00 
—11.19 

— 3.54 

— 9 65 
0. 


Dep. 


M. D. 

19.49 
35.01 
35.01 
33.04 
19.80 
16.20 
0. 


D.M.D. 

19.49 
54.50 
70.02 
68.05 
42.84 
36.00 
16.20 


N. area. 


N. 330 30' E. 
N. 76° E. 

South 

8. W. 

s. 75o w. 

8. 20o 30' W. 
W. 


+19.49 

4-15.52 

0. 

— 1.97 
—13.24 

— 3.60 
—16.20 


575.1499 
210.9150 

786.0649 












! 



S. area. 



630.1800 
761.4795 
151.6536 
347.4000 



1890.7131 
786.0649 



2)1104.6482 



10)552.3241 



This result shows that the field contains 55 acres, and 
a little more than 23 hundredths of an acre. 55.23241 

4. What is the area of a survey, of which the following are the 
field notes. 

Bearings. 

S. 46° 30' E. 
S. 51° 45' W. 

West. 
N. 56° W. 
K 33° 15' E. 
S. 74° 30' E. 



Distances. 

80 rods. 
55.16 
85.00 
110.40 
75.20 
123.80 
Ans. 104.35 acres. 
5. Required the contents and plot of a piece of land, of which the 
following are the field notes. 



Stations. 
1 

2 
3 
4 
5 
6 



Stations. 

1 


Bearings. 

5. 34° W. 


Distances. 

3.95 ch. 


O 

3 
4 
5 


S. 
S. $G%°E. 

N. b$\°E. 
N. 25° E. 


4.60 
8.14 
3.72 
6.24 


6 
7 


N. 16° W. 
JT. 65° w. 


3.50 
8.20 
Ans. 10-4. OB. 5P. 



100 


SURVEYING 


• 


6. Required the contents and plot of a piece of land, from the 


following field notes. 






Stations. 


Bearings. 


Distances. 


1 


S. 40° W. 


70 rods. 


2 


N. 45° W. 


89 


3 


N. 36° E. 


125 


4 


N. 


54 


5 


S. 81° E. 


186 


6 


S. 8° JF. 


137 


7 


W. 


130 
Ans. 207.4. 3B. 33P. 


7. Given the following bearings and distances of the several sides 


of a field, namely, 






1. 


JST. 58° JS'. 


19 ch. 


2. 


J01 6° & 


20 


3. 


& 17° TT. 


20 


4. 


W. 


20 


5. 


JST. 42° 35' JP1 


15,10 


to find the area. 




Ans. 54.9 acres. 


8. Given the following bearings and distances, namely, 


1. 


K 45° ,& 


40 ch. 


2. 


& 30° W. 


25 


3. 


S, 5° E. 


36 


4. 


W 


29.60 


5. 


K 20° E. 


31 



to find the corrected difference of latitude and departure, and the 



N. B. — In this last example, as in most others, the northings and 
southings will not exactly balance ; nor will the eastings and west- 
ings balance. This arises from inaccuracies in the data. In such 
cases ( if the errors are but trifling ) we balance off the errors. 

When a course is east or west, as the 4th in this example, some 
operators have expressed doubts, whether any correction] should be 
applied to latitude in that course. 

We reply, that errors do really exist, and, therefore, we cannot 



MENSURATION OF LANDS. 



101 



say that the course marked west, in the example, was really west, or 
not ; the probability is, that the course was not exactly due west, 
and it is therefore proper to put a correction in the latitude column, 
as shown in the following results : 

CORRECTED LATITUDES AND DEPARTURES. 



1. 


N. 


s. 


E. 


w. 


28.30 




28.30 




2. 




21.63 




12.49 


3. 




35.84 


3.16 




4. 


0.02 






29.59 


5. 


29.15 




10.62 




57.47 


57.47 


42.08 


42.08 



On inspecting these latitudes and departures, we perceive station 
5 is the most westerly point of the field, therefore to find the area, 
we will arrange these results in the following order : 



Lat. 



+29.15 
+28.30 
—21.63 
—35.84 
+00.02 



Dep. 



+ 10.62 
+28.30 
—12.49 
+ 3.16 
—29.59 



10.62 
38.92 
26.43 
29.59 
0. 



D. M. D. 



10.62 
49.54 
65.35 
56.02 
29.59 



309.5730 
1400.9820 



0.5918 



1711.1468 



1403.5205 
2007.7568 



3410.2773 
1711.1468 



2 )1699.1305 
10 )849.5652 
Ans. areas, 84.956 

9. What is the area of a survey of which the following are the 
field notes. 

From the place of beginning, N. 31° 30' W., distance 10 chains : 
thence N. 62° 45' K, 9.25 chains : thence S. 36° K, 7.60 chains : 
thence S. 45° 30' W., 10.40 chains, to the place of beginning. 

Ans. 8 T 5 / F acres. 

10. Do the following bearings and distances enclose a space ? 
If not, give an additional bearing and distance that will, then deter- 
mine the area so enclosed. 



102 





SURVEYING 




Stations. 


Bearings. 


Distances. 


1 


JS. 40° 30' E. 


31.80 ch. 


2 


JV. 54° 00' E. 


2.08 


3 


JV. 29° 15' E. 


2.21 


4 


K 28° 45' j£ 


35.35 


5 


JV. 57° 00' IF". 


21.10 



Ans. These bearings and distances do not enclose a space. A 
line run from the further extremity of the 5th to the first station 
will bear south 46° 43' W. t distance 31.21 chains, and the area thus 
enclosed will contain 92.9 acres. 

11. Do the following bearings and distances enclose a space ? If 
not, determine the additional line that will, and the area of the space 
so enclosed. 

Stations. Bearings. 

1 S. 85° 00' W. 

2 JV. 53° 30' W. 
*3 JV. 36° 30' E. 

4 JV. 22° 00' E. 

5 S. 76° 30' E. 
Ans. These bearings and distances do not enclose a space. A 

line run from the last station to the first would bear S. 13° 25' W. t 
distance 128.6 rods. Area 54.86 acres nearly. 

The operation for the area is as follows : 

We commence at station 3., for reasons that have been several 
times explained. Station 6 is the one we supplied. 



Distances. 

46.4 rods. 
46.4 " 
76.8 " 
56.0 « 
48.0 " 



Lat. 



-h 61.73 
|+ 51.92 



11.21 



*6 —125.18 



— 4.85 
H- 27.59 



Dep. 

+45^65 
+20.98 
+46.67 
—29.85 
—46.15 
—37.30 



M. D. 



45.65 
66.63 
113.30 
83.45 
37.30 
0. 



45.65 
112.28 
179.93 
196.75 
120.75 

37.30 



N. area. 



2817.9745 
5829.5776 



1029.1070 



S. area. 



2017.0153 

24629.1650 

585.6375 



9676.6590 



27231.8178 
9676.6590 



2)17555.1588 



160)8777.5794(54.86. 



* The stations marked with a * are those supplied. 



THE MERIDIAN LINE. 



103 



12. What is the 
field notes. 


area of a survey of which the 


following are the 


Stations. 

1 


Bearings. 

JST. 75° 00' 


E. 


Distances. 

54.8 rods. 


2 


iV. 20° 30' 


E. 


41.2 


< 




3 


E. 




64.8 


< 




4 


S. 33° 30' 


W. 


141.2 


c 




5 


S. 76° 00' 


W. 


64.0 


( 




6 


JV. 




36.0 


c 




7 


S. 84° 00' 


w. 


46.4 


< 




8 


JV. 53° 15' 


w. 


46.4 


t 




9 


iV. 36° 45' 


E. 


76.8 


t 




10 


2T. 22° 30' 


E. 


56.0 


f 




11 


S. 76° 45' 


E. 


48.0 


t 




12 


Si 15° 00' 


W. 


43.4 


I 




13 


S. 16° 45' 


W. 


40.5 


a 


In this survey 4 is the most easterly and 9 the 
station. The area is equal to 110A 2E. 23P. 
little, on the account of the way in which the balanc 


most westerly 
Et may vary a 
ng is done. 



CHAPTER III. 

ON THE MERIDIAN LINE AND THE VARIA- 
TION OF THE COMPASS. 



The meridian is an astronomical line, having no necessary con- 
nection with the magnetic needle. Meridians would be primary 
lines to which we would refer all surveys, if there were no such 
thing as magnetism, or a magnetic needle. 

It is only a coincidence, that the magnetic needle settles near 
the meridian, so near, that for a long time it was considered the 
meridian itself, but accurate observations have shown that the 
needle does not point rigorously north and south, but has a variation 
which is not the same at all times in the same place, therefore a 
line run by the compass is still unknown in respect to the meridian, 



104 SURVEYING. 

unless we know the variation of the compass, that is, the declination 
of the needle. 

In the year 1657 the needle at London pointed due north, since 
that time its variation has been west, previous to that time the vari- 
ation was east. 

In the Atlantic ocean, between Europe and the United States, the 
variation is from 12° to 18° west. 

The needle seems to point to the region of greatest cold, which 
is in the northern part of America, and not the north pole, and if 
this be time, if the point of minimum heat changes its position, there 
will be a corresponding change in the direction of the magnetic 
needle. 

The needle has a small annual and also a diurnal variation, cor- 
responding to the temperature of the different seasons of the year, 
and of the different times of the day, but these variations are too 
small to trouble the common operations of surveying. 

Some of the variations of the compass are regular, others irregu- 
lar ; some amount to many degrees and require a long period of 
, time, others are small in amount and require but short intervals to 
pass through all their changes. 

The daily variation consists of an oscillation eastward and west- 
ward of the mean position, and is different in different places. 
Generally the greatest oscillation eastward is between six and nine 
in the morning, and westward about one in the afternoon, gradually 
returning toward the east until eight P, M. At night it is sta- 
tionary. 

On the subject of magnetism we know nothing, beyond facts 
drawn from observation, but there is no doubt that the earth is a 
great magnet, made so by the action of the sun, and the poles of 
this great magnet are near the poles of the equator. Indeed, all 
observations made correspond to this hypothesis, for changes of the 
weather, clouds, and storms, all have an influence on the needle. 

These facts are sufficient to convince any reader that to survey 
correctly we must know the 

VARIATION OF THE COMPASS. 

As the true meridian is an astronomical line, we must find it by 
astronomical observations, and then by comparing the meridian of 



VARIATION OF THE COMPASS. 105 

the compass with it, we shall have the variation of the compass. 

When the sun is on the equator, it rises due east, and sets 
directly in the west. Should we then observe the direction of its 
center, just as it was rising or setting, at the time it had no declina- 
tion, and trace that line a short distance on the ground, we should 
then have a due east and west line. 

If from any point in that line we draw another line at right angles, 
we should then have the true meridian. 

If we now put the compass on this meridian, and make the sight- 
vanes range with it, the needle will also range with it, if there is no 
variation, but if the north point of the needle is to the west of the 
sight-vane, the variation is westerly, if to the east, easterly, and the 
number of degrees and parts of a degree that the needle deviates 
from the direction of the sight-vanes shows the amount of the 
variation. 

But it is not to be supposed that any particular observer can be 
at the points and places, where the sun is either rising or setting 
just at the time the sun is on the equator. We must have a broader 
basis, and in fact by means of the latitude of the observer and the 
declination of the sun, any observer has the means of knowing the 
precise direction in which the sun will rise or set, any day in any 
year. 

Let us suppose that the sun on a certain day, observed from a 
certain place, must have arisen S. 81° E. y but by the compass it was 
observed to rise S. 79° E. } the variation of the compass was there- 
fore 2° west. 

These observations are called taking an azimuth. Azimuths are 
often taken at sea to determine the variation of the compass. 

On land, however, the horizon is rarely visible, and very few obser- 
vations on sun rise or sun set can be made, besides there are other 
objections arising from atmospherical refraction ; it is therefore best, 
most convenient, and more conducive to accuracy, to take the sun 
when up 10, 15 or 25° above the horizon, and observe its direction 
per compass, and compare the result to the computed bearing for 
the same moment, and if the two results agree the compass has no 
variation ; if they disagree the amount of such disagreement is the 
amount of the variation of the compass. 



106 



SURVEYING 



By means of spherical trigonometry the true bearing of the sun 
can be determined at any time, on the supposition that the observer 
knows his latitude, the declination of the sun, and its altitude ; these 
three conditions furnish a triangle like PZS. 

The altitude subtracted 
from 90°, gives ZS, the lati- 
tude from 90°, gives ZP, and 
PS is found by adding or 
subtracting the sun's declina- 
tion to 90°, according as it is 
north or south. 

ZS is co-altitude, ZP is 
co-latitude, and PS'is the sun's 
polar distance ; the angle PZS 
is required, and it can be 
found by the following rule. 

1 . Add the three sides of the triangle together and take the half sum. 
From the half sum, subtract the sun's polar distance, thus finding the 
remainder. 

2. Add the sin-complement of the co-altitude, the sin-complement of 
the co-latitude, the sine of the half sum, and the sine of the remainder. 
The sum of these four logarithms divided by 2, will be the cosine of 
half the azimuth angle. 

"N. B. This rule is the application of equations on page 204 
Robinson's Geometry. The sin-complement is the logarithmic sine 
of an arc, subtracted from 10. 




EXAMPLES. 

In latitude 39° 6' 20" north, when the sun's declination was 12° 
3' 10" north, the true altitude of the sun's center was observed to 
be 30° 10' 40", rising. What was the true bearing of the sun, or its 
azimuth ? 

90° 90 90 

Lat. 39 6 20 Alt. 30. 10. 40 Dec. 12° 3' 10 

co-Lat. 50 53 40 co-Alt. 59. 49. 20 PJ) 77. 56 50 



VARIATION OF THE COMPASS. 107 



P. D. 


77° 56. 50 




co-Lat. 


50. 53. 40 sin. com. 


0.110146 


co- Alt. 


59. 49. 20 sin. com. 


0.063295 



2)188 39 50 

94 19 55 
77 56 50 


sin. 
sin. 

cosine 


9.997758 


16 23 5 


9.450376 




2)19.622575 


49° 38' 30" 


9.811287 



is. 

Rem. 



Bearing, 99° 17' from the north, or 80° 43' from the 
south. 

If at the time of taking the altitude of the sun, another observer 
had taken its bearing by the compass, and found it to be S. 80° 43' 
JE., then the compass would have no variation, and whatever it dif- 
fered from that would be the amount of variation. 

If a line were run along the ground, direct toward the center of 
the sun, at the time the altitude was taken, and sufficiently marked, 
that would be a standing line of known direction ; and if from any 
point in that line, we could draw another line, making an angle with 
it of 99° 17' on the north, or 80° 43' on the south, such a line 
definitely marked, would be a permanent meridian line, for all time 
to come ; on which we could at any time place a compass, and 
observe its variation. 

Let AS be the line toward the sun, 
along the ground, AE a fine due east, and 
Mm a true meridian line. The angle SAM 
must equal 9° 17'. 

To make that angle, take.^^, one chain 
or 100 links ; from S, draw the line SE 
at right angles to AS, by means of a 
surveyor's cross.* 

From £take SIS, of such a value as will make SAE 9° 17', 
which is determined by trigonometry ; as follows, 

* Surveyor's cross is nothing more than a pair of sight vanes, set at right 
angles with each other, for the purpose of making right angles. 






SURVEYING. 




As 100. 


: tf^Rad. : tan. 9° 17' 




Whence 


„„ 100. (tan. 9° 17') 


9.213405 

2 



108 



SE=16.S47 links. 1.213405 

That is, from S measure off 16 and a little more than -J- of a link, 
and there is the point E. A line drawn from A to E, is a due east 
and west line. 

If we put the surveyor's cross on this line, at any point as A, and 
range one branch of it along the line AE, the other branch will 
mark out the line Mm, a true meridian, if everything has been done 
to accuracy. 

In the afternoon, or some other day, another meridian may in 
like manner be drawn near this one, and if they are both true meri- 
dians, they will be parallel. If not parallel, other observations 
should be made until some two or three are obtained, that are 
parallel or very nearly so ; and the mean direction then, may be 
regarded as the true meridian, 

A true meridian will always be a test line for a compass ; and by 
placing any compass upon it, the declination* of the needle can be 
determined. 

Again. In the triangle PZS, if we compute the angle ZPS (as 
is done on page 211 Robinson's Geometry), we shall have the sun's 
distance from the meridian or the apparent time ; then if we have a 
time piece that can be relied upon, for three or four hours, we can 
determine the time within a few seconds, when the sun will be on 
the meridian. A line at that time, run direct toward the center of 
the sun, will define the meridian. 

The objections to these methods are, 

1. The sun is a large body, and its center cannot be exactly 
defined. 

2. The sun changes position so rapidly that, unless we are in an 
observatory, where every thing is prepared and in order, it is diffi- 
cult to get observation upon it. 

* Declination of the needle, in common language, is called the variation of the 
compass ; and, as a general thing, we adhere to common language. 



VARIATION OF THE COMPASS. 109 

3. The sun is so bright an object that it cannot be viewed with- 
out prepared glasses. 

4. The majority of persons that have been, and probably will be 
practical surveyors, have not the instruments to take altitudes of the 
sun, and they are not and cannot be at home in astronomical obser- 
vations and computations. 

Some of these objections are deserving of little respect, and others 
can be partially removed. 

For instance, if the sun is too large, and too brilliant to be accur- 
ately and deliberately observed, we can take the planet Yenus, Jupiter, 
or Saturn, and, by proper observations, determine their directions, 
during the twilight of evening, when we can see the planet distinctly, 
and at the same time that other objects are sufficiently distinct to run 
lines.* 

But the method most known and most in favor among practical 
men, is that of taking the direction of the north star. 

The north star is a star of the second magnitude (Polaris), whose 
right ascension, Jan. 1, 1851, was lh 5m 18s ( at present increas- 
ing at the rate of 17s 71 per annum ), and declination was then 88° 
30' 55", with an annual increase of 19"8, it is therefore, but 1° 29' 
5" from the pole, and it is called the pole star or north star because 
it is so near the pole. 

If the star were situated directly at the polar point, a line toward 
it would be the true meridian line, but being 1° 29' 5" distant, the 
star apparently makes a circle round the pole in a siderial day, 
making two transits across the meridian, one above and the other 
below the pole, — a direction to it, at these times, would be a true 
meridian line. 

To find these times, subtract the right ascension of the sun from the 
right ascension of the star ; increasing the latter by 24h, to render 
the subtraction possible, when necessary. 

* For example, in the year 1853, from the 25th of July to the 5th of August, 
the planet Jupiter will pass the meridian in the evening twilight. On the first 
of August, Jupiter will pass the meridian of New York, at 8h 11m 53s, and it 
will pass the meridian of Cincinnati, at 8h 11 39s, mean local time ; and, of 
course, whoever is able to designate that time within a few seconds, and is also 
prepared to mark the direction of the planet, will have a true meridian line. 

The moon is not a good object for this purpose; it changes its place too rapidly. 



H. 


M. 


s. 


25 


6 





6 


41 


16 


18 


24 44 



110 SURVEYING. 

The difference will be the time of the upper transit, and llh and 
59 minutes from that time will be the time of the lower transit. 
The right ascension of the sun is to be found in the Nautical Alma- 
nacs, for every day in the year ; and it is nearly the same, for the 
same day, in every year. 

For example. At what times will the north star make its transits 
over the meridian on the first day of July, 1853. 



■* JR. A+24h 
O B. A. - 



This result shows that the upper transit will occur about 6h 24m, 
in the morning of the 2d of July. I say about, because I took the 
sun's right ascension for the morning of July 1, and from that time 
to 6, next morning, is 1 8 hours : and during this time the right as- 
cension of the sun will increase full 3 minutes, — therefore the 
upper transit will take place 6h 21m in the "morning, and the pre- 
vious lower transit 1 lh 59m previous, or at 6h 22m, evening. 

But neither of these transits will be visible, as they both occur in 
broad day light, from any place where the north star is ever dis- 
tinctly visible. 

In summer, then, when most surveying is done, the meridian 
transits of the north star are not visible, nor is this important : for 
the transits are seldom used, by reason of two objections : 

1. The star changes its direction most rapidly while passing the 
meridian. 

2. Observers, generally, have not the means of knowing the time 
to sufficient accuracy.* 

To obviate these objections, observations may be taken on the 
star at its greatest elongations ; for, about those points and for full 

* Note. — Very few persons consider that their clocks and watches, however 
good and valuable, do not give the exact time, but only approximations to the 
time. 

For any astronomical purpose, like the one under investigation, the charac- 
ter of the time piece should be well tested — its rate of motion known — and 
its errors established by astronomical observations. 



VARIATION OF THE COMPASS. 



Ill 



15 minutes before and after, the star does not visibly change its 
direction ; hence the observer has a sufficient interval to be deliber- 
ate, and he can be sufficiently exact as to time without any extra 
trouble. 

The following tables show the times of the greatest eastern and 
western elongations, which occur in the night season. These tables 
are not perpetual, but they will serve without correction for 20 years 
or more to come. 



EASTERN ELONGATIONS 



Days. 


April. 
H. M. 


May. 
H. M. 


June. 


July. 


August. 


Sept. 
H. M. 


H. M. 


H. M. 


H. M. 


1 


18 18 


16 26 


14 24 


12 20 


10 16 


8 20 


7 


17 56 


16 03 


14 00 


11 55 


9 53 


7 58 


13 


17 34 


15 40 


13 35 


11 31 


9 30 


7 36 


19 


17 12 


15 17 


13 10 


11 07 


9 08 


7 15 


25 


16 49 


14 53 


12 45 


10 43 


8 45 


6 53 



WESTERN ELONGATIONS. 



Days. 


Oct. 


Nov. 


Dec. 


Jan. | Feb. 


March. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


1 


18 18 


16 22 


14 19 


12 02 


9 50 


8 01 


7 


17 56 


15 59 


13 53 


11 36 


9 26 


7 38 


13 


17 34 


15 35 


13 27 


11 10 


9 02 


7 16 


19 


17 12 


15 10 


13 00 


10 44 


8 39 


6 54 


25 


16 49 


14 45 


12 34 


10 18 


8 16 


6 33 



It will be observed that these times are astronomical ; the day 
commencing at noon, and 12h 40 means 40m after midnight, etc. 

Now, admitting that the direction of the star can be observed, the 
next step is to find how much that direction deviates from the meri- 
dian — and this is a problem in spherical trigonometry. 

A great circle passing through the zenith of the observer to the 
star, when the star is at one of its greatest elongations will touch the 
apparent small circle made by the apparent revolution of the star 
about the pole, and will therefore, with the star's polar distance, form 
a right angle — and we shall have a right angled spherical triangle, 
of which the observer's co-latitude is the hypotenuse, the star's polar 
distance one side, and the angle opposite to this side is the angle 
required. 



112 



SURVEYING. 



EXAMPLE. 

What will be the bearing of the north star observed from latitude 
42° JV. in the year 1860, when the star's polar distance will be 
1°26'12"? Ans. 1° 56'. 

As cos. Lat. 42° 9.871073 

is to radius 10.000000 

So is sin. 1° 26' 12" - - - - 8.399183 

To sin. 1°56' 8.628110 

In this manner the following table was computed. The mean 
angle only is put down, being computed for the first of July in each 
year. 

AZIMUTH TABLE. 



Years. 


Lat. 30° 


Lat. 35° 


Lat. 40° 


Lat. 45° 


Lat. 50° 


Azimuth. 


Azimuth. 

1°48'21" 


Azimuth. 


Azimuth. 


Azimuth. 


1852 


1° 42' 30" 


1° 55' 52" 


2° 5' 32" 


2° 18' 5" 


1854 


104^45- 


1°47'39" 


1°55' 2" 


2° 4' 30" 


2° 17' 6" 


1856 


1°41' 2" 


1° 46' 49" 


1° 54' 12" 


2° 3' 44" 


2° 16' 9" 


1858 


1° 40' 27" 


1°46' 11" 


1° 53' 30" 


2° 3' 2" 


2° 15' 12" 


1860 


1° 39' 43" 


1°45'24" 


1°52'32" 


2° 2' 4' 


2° 14' 16" 


1862 


1° 38' 50" 


1°44'29" 


1°51'44" 


1°1' 2" 


2° 13' 18" 



This table is given for those who may wish to use it, but we 
would recommend each observer to follow the example which pre- 
cedes the table, and compute the azimuth corresponding to his 
latitude and time. 



THE PRACTICAL DIFFICULTY. 

The north star is not brilliant, it cannot be seen until it is so dark 
that all minute terrestrial objects are totally invisible, it is therefore 
difficult to draw a line and accurately mark it. All these night 
operations are, at best, perplexing and inaccurate^ yet, by the means 
of fights and artificers, lines can be drawn. 

If the observer have a theodolite and an assistant, there will be no 
difficulty. Let them be at the place from which they wish to take 
the observation in time, adjust the instrument and direct the telescope 
to the north star. Now sufficient lio-ht must be reflected into the 

o 

telescope to enable the observer to see the cross hairs, and this may 
be done by the assistant holding a light before a stiff sheet of white 



TO SURVEY WITHOUT A COMPASS. H3 

paper, so as to throw the reflected light from the paper into the 
telescope, or this may be done by means of a stand to hold both the 
light and the paper. 

When the vertical spider's line becomes visible, let the star be 
brought directly upon it, and if it is near the time of greatest elon- 
gation it will appear to remain so, for some time. But if the star 
has not reached it greatest elongation, it will move from the line 
more to the east, if the elongation is easterly, and more to the west, 
if westerly. 

The telescope must be continually directed to the star, by means 
of the tangent screw of the horizontal plate, but for some time the 
spider line and star will coincide without moving the screw, and 
then the star will depart from the line in the contrary direction to its 
former motion, but the telescope must no longer follow the star, its 
position will now show the direction to the star, when the star had 
its greatest elongation, and thus it should be left until morning. 

In the morning, carefully range and mark a line through the 
telescope. 

If we now make an angle with this line equal to the azimuth, by 
means of the theodolite, or by means of measuring a triangle as 
explained in the former part of this chapter, and mark this new line 
either to the right or left, as the case may require, we shall then 
have a permanent meridian Kne for all future use. 

By placing a compass on any well defined and true meridian we 
can determine its variation by simple observation. 

If we have not a theodolite, we can obtain a tolerably accurate 
direction to the north star by means of illuminated plumb lines sus- 
pended in vessels of water, so placed as to range to it. 



CHAPTER IV. 

TO SURVEY WITHOUT A COMPASS. 

The inquiry is sometimes made, whether lands could be surveyed 

without a compass ; we reply in the affirmative. The compass is 

only a convenience, and if it had never been discovered, it is probable 
8 



114 



SURVEYING, 



that surveys would have been more accurately made. Too much 
reliance has been placed on the accuracy of the compass, and in 
consequence little attention has been paid to denning any astronomi- 
cal lines. 

Were it not for the compass, it is probable, that every country- 
town, and even every large land holder, would have meridian lines 
well defined about his premises. 

Having a meridian line to start upon, we can find angles and 
define the position of fines very accurately by means of a 



CIRCUMFERENTOR. 

The circumferentor consists of a horizontal circular plate divided 
into 360 degrees, over which an index bar, or another circular plate, 
is made to revolve. This index bar carries sight-vanes or a telescope. 
The index bar or the revolving circular plate also carries a vernier 
scale, which will enable the operator to make an angle to one minute 
of a degree. The whole instrument is placed on a tripod, and by 
the aid of spirit levels attached to the lower plate, the horizontal 
position is attained with a sufficient degree of accuracy. 

The figure before 
us, represents the 
essential parts of a 
circumferentor. JVS 
is considered as the 
primative or meri- 
dian line, and AB is 
the index bar, which 
turns horizontally on 
the common center. 

Vernier scales are 
fitted into the index 
bar, and revolve over 
the graduated arc. 
At A and B are 
openings, to receive 
cross hairs or a telescope. 

When a vertical semicircle is made to revolve vertically through 




TO SURVEY WITHOUT A COMPASS. 115 

the plane AB, and the diameter of that circle a telescope, then we 
have all the essentials of a theodolite. 

To most theodolites, a magnetic needle is attached, but the mag- 
netic needle is, properly speaking, no part of the instrument. 

To show the manner of finding the direction between two given 
points, by means of the circumferentor, we propose the following 
problem. 

Mr. T. H. Jones wishes me to run the east line of his lot, in the 
town of A, and give the true bearing, the corners being known. In the 
public square of the town, about one and a quarter miles distant, a 
meridian line has been established. 

Let Mm be the established meridian in 
the public square, and OH the direction of 
the line required. 

Place the circumferentor on the meri- 
dian line Mm, so that NS of the instru- 
ment will coincide with it, the center of 
the instrument being at a in a road. 

The general direction of the road is a b, 
and the index bar AB is made to revolve over the plate, which is 
firmly fixed, until the index bar or sight vanes point out the line 
ab. The line is run by means of ranging objects ; such as flag staffs, 
if the line is long ; or if short, by sending on a flag, and stationing 
it at b. 

Now clasp the index bar on the plate, by the clamp screw under 
it (made for the purpose). Leave a flag at a ; take the instrument 
to b, and there place it, so that the sight vanes will range back to a; 
then the position of JV$ on the instrument will show a meridian line 
through that point. 

Here the road bends a little, unclamp the index (being careful 
that J¥S rigidly retains its position), and direct it to the general 
direction of the road be. Mark the point c, by an object as before, 
and mark some other point, so as to secure the line (the other point 
may or may not be b). • Now clamp the index again, and remove the 
instrument to c. Place the instrument firmly as before, and make 
the index range along the line be; the line JVSof the instrument, 




116 SURVEYING. 

will mark out a meridian line at the point c, and tlus we can transfer 
the meridian line Mm to any other point whatever. 

Thus we may go to any point d, in the given line ; no matter, 
theoretically speaking, how many angles we have made during the 
traverse. Placing the instrument at d, with its index to range along 
the last line, the line NS of the instrument gives the meridian if ' m'. 

Now unclamp the instrument, and direct its index along the 
required line GIT; the position of the index, on the graduated plate, 
will give the angle from the north, which, by means of the vernier, 
can be determined with great exactness. 

In this manner we may go to any point, and place a meridian 
there, and then run any required line whatever ; therefore, we can 
survey any field, farm, or tract of land, without a compass, if we 
have a circumferentor, and a meridian line. 

It would not be safe to transfer meridians, as we have just done, 
over any very great extent of country, for at every angle, small 
errors might be made, and the accumulation of many small errors 
may produce too great inaccuracies to be tolerated or overlooked. 
When using the magnetic needle, no errors accumulate, for every 
setting of the compass is primary, and independent of every other. 

Therefore, in case no compasses were in existence, primary meridi- 
ans, astronomically established, would be necessary in every town ; 
and it would be better to have several of them in the same town. 

From the foregoing illustrations, we perceive that surveying can 
be done, and well done, without a compass, yet the compass is an 
inestimable blessing to mankind ; for it is the only index to direction 
over the wild waste of waters, when the heavens are obscured, and 
no mariner would dare brave the ocean without it. 



CHAPTER V. 

ORIGINAL AND SUBSEQUENT SURVEYS.— 

DIFFICULTIES AND DUTIES OF 

A SURVEYOR. 

In this country, lands were ceded to States, or sold to companies 
in large tracts, without any definite surveys; the boundaries 



SURVEYS AND SURVEYORS. 117 

described, were mountains, rivers, or a certain number of miles 
along the shores of a lake, and then a certain number of miles back. 
The land companies hired surveyors from time to time, to survey 
off their lands, into lots of 100, 200, and 500 acres ; and wherever 
these surveyors left monuments for the corner of lots, established the 
corners for all time to come, whether correctly placed or not. 

These surveys were very loose and inaccurate ; it could not be 
otherwise, for a company of surveyors would frequently run 15 
miles in a day ; when to run a line accurately, and measure it, four 
miles is a good day's work. 

But, notwithstanding inaccuracies, these surveys are legal and 
cannot be changed ; " thou shalt not move thy neighbor's ancient 
land mark," and it is right it should be so ; for any attempt at cor- 
rection, would create more trouble, confusion, and injustice, than 
it could remedy. 

Lots originally sold for 100 acres in the state of New York, 
generally contain from 101 to 106 acres, in consequence of the orig- 
inal surveyors having directions to have their lots hold out. Where 
the lots thus overrun in one portion of the tract, they fall short on 
another, for the surveyors were probably desirous to show to the 
company, that their grant actually contained as much land as was 
anticipated. 

The author surveyed one of these lots, that originally sold for 100 
acres, and found that it contained but a little over 76 acres. 

Some of the companies had their grants laid off into townships 
6 miles square, or 6 by 8 miles ; then each township into four sec- 
tions, each section divided off into lots, and the lots numbered, 
generally beginning at the south-west corner. 

The description of the lots in the deeds given, were very loose 
and indefinite, stating the township, section, and number of the lot, 
containing 100 acres, " be the same more or less," and in some lots 
it was more, and in other lots it was less. 

As we before remarked, any land mark to the corner of a lot laid 
down by these original surveyors, must remain ; subsequent sur- 
veyors can straighten lines between point and point, and decide what 
the true courses are, and how many acres the lot contains. 

When a surveyor is called to survey any farm or estate that has 



118 SURVEYING. 

been previously surveyed, he must find some corner as a place of 
commencing, and from thence run a random line, as near the true 
line as his judgment permits ; and if he strikes another corner he has 
run the true course, if not, he corrects his course, as taught in chap- 
ter II. Thus, he must go round the field from corner to corner. 
He has a right to establish corners only where no corners are to be 
found, and no evidence can be obtained as to the existence and 
locality of a former land mark. 

It may be the case, that a surveyor is called to survey a lot where 
no corners are to be found. If a fence or line exists, which has 
been the undisputed boundary for a long time, that boundary can- 
not be changed, and the surveyor must establish a corner by ranging 
some other line to meet the first. Sometimes corners may be found 
to some neighboring lot, from which lines can be run, to establish a 
corner to the lot we wish to survey. 

Lines of lots in the same town, are generally parallel, and a sur- 
veyor who offers his services to the public, must make himself 
acquainted with the general directions of the lines of lots, over that 
section of country where his services are required. 

When a surveyor is called to divide a piece of land, he is then an 
original surveyor, and not liable to be embarrassed by old lines and 
old traditions, he has then only his mathematical problem before 
him. 

Owing to the inaccuracies of original surveys, and the impossi- 
bility of leaving proper land marks, in consequence of the great 
haste in which lands were originally surveyed ; great confusion has 
followed, in some sections of our country, in respect to lines, and it 
has been no uncommon thing to have whole neighborhoods at vari- 
ance, if not in law, in reference to the boundaries of their lands. 

In cases of this kind, one, and then another of the disaffected, 
have successively employed surveyors, and surveyors thus employed, 
are apt to act the part of advocates, rather than arbitrators, and 
survey too much according to the direction of their employer ; but 
all such efforts to settle difficulties, but aggravate them more and 
more. 

On the contrary, however, if the surveyor clearly understands his 
duties, and can rise above being a special advocate for any one of 
the parties concerned, he can do more than judges or juries to restore 



SURVEYS AND SURVEYOR'S. H9 

harmony and peace. To illustrate these views, and possibly to give 
some valuable instruction to some readers, we give a history of a case 
of this kind, which occured in the year 1837, in the county of 
Ontario, in the State of New York. A tract of land consisting of 
about 670 acres, of an irregular shape, was divided on paper into 
five equal parts, and sold to five different individuals. 

The whole 670 acres was bounded by four lines, no two of them 
were equal, and neither of the angles was a right angle. The 
largest boundary line could not be directly measured on account of 
an impassable ravine ; and the banks of this ravine was so thickly set 
with hemlocks, that it was impossible even to sight across. 

In consequence of the irregular shape of the whole, and the 
impossibility of directly measuring the principal boundary, they had 
never been able to agree on their division lines. 

Each one imagined that his neighbor was inclined to crowd upon 
him, and although permanent fences were desirable, none could be 
made until lines were agreed upon. They had employed several 
surveyors, but they had not been able to agree on their divisions. 
In this state of things a surveyor was called upon to go and make 
a division of this land, but the difficulties of so doing were carefully 
concealed from him. 

When he arrived on the ground, ready for operations, the whole 
neighborhood was present, and by unmistakable signs he soon 
learned that an unusual degree of interest was taken in the survey. 

He also found that the chief difficulty arose from not being able 
to measure the line CD. All the corners, A, B, 0, and D, were 
established. The surveyor commenced at C to run a random line 
as near CD as possible. After going a few chains, he came to the 
bank of the ravine at F, where it was impossible to pass or sight 
across. Driving a stake at F, he took a direction FH along the 
bank of the ravine, carefully 
noting the angle, and meas- 
uring the line to H, a point 
where objects were clearly to 
be seen on the other side of the 
ravine. The surveyor then 
sent a man over with a flag, 
stationing his staff, first at K t 




120 SURVEYING. 

then at L, carefully noting the direction of each, and being careful 
to have the angle KHL greater than 30°. He then passed over and 
set the compass at K, took the direction of KL and measured it. 
Having now KL one side, and all the angles of the triangle BKL, 
he computed HL. He now set the compass at L and took a definite 
direction Lm ; this definite direction gave him the angle HLm, and 
he now had all the angles of the quadrilateral LmFH, and two of 
its sides. Whence he computed the exact distance to m, to strike 
the line CF produced. 

He measured that distance and drove a stake at m, and computed 
mF. The company now ran through the random line, driving 
stakes at the end of every third chain, and the random line came 
out within a few feet of the established corner at D. The surveyor 
measured the perpendicular distance to D, and corrected the course 
by the rule in chapter II. He also computed how far each stake 
that had been placed on the random line must be moved to transfer 
it to the true line ; this, the reader will perceive, was done by propor- 
tional triangles. 

At D he set the compass, and carefully noted the course and dis- 
tance to A, He then returned to C, taking care not to pass along the 
line AB. 

At O he set the compass, and carefully noted the course and 
distance to B. He now computed the course and distance from 
B to A. The line lay in the open fields, over tolerably smooth ground, 
and it could be directly and accurately measured. 

The surveyor now called all the parties interested, including the 
sour and the belligerent, and told them that the distance from B to A 
was a certain number of chains and finks, and that they would now 
measure it, and if they found it to correspond without any material 
error, they must then be convinced that he had obtained the true 
length of CD, and that he could then divide the land into five equal 
parts, as required. 

To this test they all cheerfully assented ; the line was measured 
and corresponded to the computation within three finks ; all parties 
were satisfied, and thus ended a neighborhood quarrel of six years' 
standing. 

Previous surveyors commenced at the point D and run DA, AB, 



SURVEYS AND SURVEYORS. 121 

and BC, and then computed CD. This was more direct, simple, and 
proper, than the method just described, but it left no test behind 
it, and it is vain to expect that the mass of men will receive theoreti- 
cal computation as actual measurement. 

Here, and in most other cases that involve contention, the surveyor 
must not only convince himself that the survey is correctly made, 
but he must, if possible, show others that his conclusions are not 
only right, but cannot be wrong ; hence judicious surveyors must 
often measure lines, where there is no mathematical necessity for so 
doing. 

The next duty of this surveyor was to divide the land into five 
equal parts. Each one had previously purchased his part, and he 
knew its locality, but not his exact boundary line on the division. 

As CD was not parallel to AB, and AD not exactly parallel to 
CB, to divide this mathematically exact was a problem of considerable 
difficulty, and this will be explained in the next chapter ; but practi- 
cally we need not apply all mathematical rigor, the surveyor can 
divide this more strictly conformable to justice without, than with 
the mathematical rigor. 

The persons who had the two most eastern lots, had the worthless 
part of the land in the ravine, and of course if any one had an excess 
of area it should be those. 

To find where or nearly where the divisions come, divide the line 
AB into five parts and suppose Q- one of those parts. Now BO is 
not quite long enough, because the field is a little narrower at this 
end than at the other ; the surveyor took a distance B G a few links 
greater than one fifth of AB, and from that point run a line in a 
medium direction between B C and AD, and then computed its area, 
the result would show whether the area was too great or too small, 
and if it were within a very small fraction of the area required, the 
line is left as the true one, otherwise it is moved as the case requires. 
In the same manner the other division lines were run. 

UNITED STATES* LANDS. 

Soon after the organization of the present government, several of 
the States ceded to the United States large tracts of unoccupied 
land, and these, with other lands, since acquired by treaty and pur- 
chase, constitute what is called the public lands. 



122 SURVEYING. 

Previous to 1802, there was no general plan for surveying the 
public lands, or in fact, no surveys were made, and when grants 
were made the titles often conflicted with each other, and in some 
cases different grants covered the same premises. 

In the year 1802, Colonel I. Mansfield, then Surveyor General 
of the north-western territory, adopted the following method : 

Through the middle, or about the middle of the tract to be sur- 
veyed, a meridian is to be run, called the principal meridian. At 
right angles to this, and near the middle of it, an east and west line 
is to be run, and called the principal parallel. 

Other meridians are to be run, six miles distant from the prin- 
cipal meridian, both east and west. 

Also, parallels of latitude are to be run, six miles from the prin- 
cipal parallel, both north and south. 

When this was done ( and it has been on all the public lands 
east of the Mississippi river ), the whole country is divided into 
squares, six miles on a side, called townships. 

Each township contains 36 square miles. Each square mile is 
called a section, and it contains 640 acres. Sections are divided into 
half sections, quarter sections, and eighths. But these divisions are 
only made on paper. 

When a person makes a purchase of a half or quarter section, it 
is supposed that he will find it himself, or employ a surveyor to 
mark it out. 

Townships which lie along a meridian, are called a range, and 
numbered to distinguish them from each other. 

Sections are regularly numbered in every township, and to desig- 
nate any particular one, we say, section 13, in township number 4 
north, in range 3 east. 

This shows that the third range of townships east of the princi- 
pal meridian, in township No. 4 north of the principal parallel, 
is the township, and the thirteenth section of this township is the one 
sought. 

Not more than ten townships north or south of a principal par- 
allel should be drawn, before a new principal parallel should be 
designated, and new measures made between meridians : because 
meridians tend toward the pole, and the north lines of townships 



DIVISION OF LANDS 



123 



will be theoretically shorter than south lines, if the meridians are 
run by the compass. 

"Where the public lands extend to rivers and lakes, there will be 
fractional townships along the shores. 

Where the locality of a particular number is found to be occupied 
by a lake or pond, the sale is void. 



CHAPTER VI. 

METHODS OF SURVEYING IRREGULAR 
FIGURES AND OF DIVIDING LANDS. 

Farms and tracts of lands, wholly or 
partially bounded by water, as represented 
in the figure before us, are surveyed and 
there areas determined by drawing right 
lines within the tract as near the real 
boundaries as possible, and from these right 
lines, at equal hUervals, measuring the off- 
sets to the real boundary. These off-sets 
form the parallel sides of trapezoids, and 
as they are all equally distant from each 
other, the computation of the areas they 
occupy will be very easy. A summary 
rule for finding the united area of all these trapezoids that are 
bounded by one line, is to be found in Prob. VIII, Mensuration. 
The area of the right fined figure ABCDEFG, is found as directed 
in Chapter IV, to which add the area of all the trapezoids, and we 
shall have the area of the whole. 

We have now investigated every possible case of computing areas, 
and we are now prepared to divide them. Commencing with the 
most simple case of the most simple figure, the triangle, or rather 
the figure that has the least number of sides. 




124 SURVEYING. 

PROBLEM I. 

To divide a triangle into two parts, having a given ratio of m to n. 

Case 1. By a line drawn from one angle 
to its opposite side. 

Let ABC represent the triangle; divide 
its base into two parts, corresponding to the 
given ratio, and let AD be one of the parts; 
then we shall have the following proportion. 

AD : AB : : m : m-\-n 




m 




Whence, AD=-^(AB) and ££=~(A1S) 

Now the two parts are numerically known, and are to each other as 
m to n. Triangles, having the the same altitudes, are to one 
another as their bases. Therefore, ADO : CDB : : m : n as re- 
quired. 

Case 2. By a line parallel to one of its 
sides. 

Let DE divide the triangle as required, 
and as similar triangles are to one another 
as the squares of their homologous sides, 
therefore : 

(AB) 2 : (AD) 2 : : m+n : m 

Whence, AD^ABa/^L. 
m-\-n 

Which shows that if we have the numerical value of AB, and of 
n and m, we can find that of AD, and from D draw DE parallel 
to BC, and the triangle is divided as required. 

Case 3. By a line parallel to a given line, or by a line running in 
a given direction. 

To make this case clear, we commence by giving a definite ex- 
ample : 

There is a triangular piece of land, from one of the angular points, 
A, one line runs JV. 25° W., distance 12 chains; another from the same 
point runs JV. 42° E., distance 15 chains. It is required to divide this 



DIVISION OF LANDS. 



125 



triangle into two parts in the ratio 2 to 3, by a line running due east 
and west. 

Let ABO be the given triangle, and B'O' 
the required division line. It is required to 
find the numerical value of A C or AB', to 
make the area AB' C § of the area ABO. 

Let b represent the side of the triangle 
opposite B, and c the side opposite 0. 

LetAC'=x. As AC and, OB' have defi- 
nite directions, the angle AO'B' is given, 
also AB'O' is given. A C"J3'=48°, AB ' 
=65°, BAO=W°. 

In the triangle AB' C we have 

sin. 65° : x : : sin. 48° : AB' 




Whence AB'= 



sin. 48° 



sin. 65° 
Now, by Prob. Ill, Mens., area ABO=\bc sin. A. 

-■', /sin. 48° \ 



Also, 



(i) 



x 2 sin. A 



By the conditions of our problem, we have the following pro- 
portion. 

'sin. 48°\ 



I be sin. A 



< 



5: 2 



Or, 



sin. 48° 

be : = — wfoX 2 : : 5 



2 



(2) 



sin. 65°" 

We may here stop, and make the problem general. 

If B' O is given in direction, the angles B' and C will be given. 

We now require the division of the triangle ABO into two parts, 
in the ratio of m to n by a line opposite to the angle A, running in 
a given direction. 

Represent the sides of the given triangle adjacent the angle A 
by b and c, b extending from A to C, and c extending from A to B. 

Put x= the distance from A to the division fine, on the side OA. 

Then, by the preceding proportion we have, 



126 SURVEYING. 

T sin. C 

he : - t5>% 2 : : m-\-n : m 

sin. hi * 



sin. 



.,... / w- \i/fcsin. J3'\i 

Whence, »=( — j— ) a ( ^ — 777 F 

\w-(-w/ \ sin. (7' / 

Observe that x is opposite the angle B', the sine of which stands 
in the numerator of the second fraction. Had x represented AB', 
sin C would have been the numerator. 

Drawing out the result for (2), we find that 
5 sin 48° . 

n . /72 sin 65° n ... , . 

Or, a:=/V — - — r^r- =9.446 chams. 

sm 48° 

Case 4. By a line that shall pass through a given point within 
the triangle. 

A point in a triangle cannot be given, unless the perpendicular 
distances from that point to the sides are given, and if these per- 
pendicular distances are given, then we can readily find the three 
distances from the angular points of the triangle, and the angles 
which these lines make with the sides 
of the triangle are known. For instance, 
if the point P, in the triangle AB 0, is 
known, PR and PT are known, and 
all the angles of the quadrilateral 
ATPR are known. These are suffi- 
cient data to compute the line AP, and 
the angles BAP and TAP. 

Remark. — We may now require a triangle to be cut off, by a line running 
through P, which shall contain any definite portion of the triangle ABC t not 
involving an impossibility. 

For instance, if the point P is near the center of the triangle, it would not 
do to require us to cut off a tenth part of the triangle, or any smaller portion, 
for it would be impossible to do so. When it is required to cut off a very small 
portion of the whole triangle, the point P must be near one of the sides, or 
near one of the angular points. Sometimes the required quantity can be cut 
off from one angular point, sometimes from another, and sometimes from all 
three. 

Let us now require one-third of the triangle cut off, by a line passing through 




DIVISION OF LANDS. 127 

P, taking the angular point A, and let EF be that line. We are to determine the 
value of AF. 

Iii the triangle AB C, the angles A, B, and C, are known, and the 
sides opposite to them, a, b and c, are also known. AP is known, and 
call it h. Put the angle EAP=p, PAF=q. Then, A=p+q. 
Put AF=x, AE=y. 

By Prob. III. Mens. area ABC=^bc sin. A 
Also " " area AEF=%xy sin. A 

By the conditions of the problem, 

\xy sin. A=^bc sin. A 
Whence, 3xy=bc (1) 

The triangle AFE consists of two parts, AFP, APE ; therefore, 
\hx sin. q-\-^hy sin. p=^xy sin. A 

r\ i • X V sin. A /0 x 

Or x sm. ^-j-y sm. p= (2) 

h 
If we had required the nth. part of the triangle ABC, in place of 
the 3rd part, equation (1) would have been nxy=bc. 

Making this supposition, to make the problem more general, we 

have#y= - and y= — By the aid of these last two equations, 
n nx 

(2) becomes 

, he sin. p be sin. A 

x sm. q-f- = 

nx nh 

~ be sin. A be sin. p 

Or, a 2 — __ x=— — r—L ,„* 

rcA sm. q ?z sin. </ I* 3 ,/ 

Whence, x- hc sin ' ^ db/ 52c2sin - 2 ^ _ ksin -A i 
2w^sin.5- \4tt 2 A 2 sin. 2 <? nsin. qj 

In case n is for^e, that is the part to be cut off small, the value 
of x may be imaginary, corresponding to the preceding remark. 

Case 5. When the given point is on one side of the triangle. 

The two parts must be equal, or one 
of them will be less than half of the 
whole. 

We always compute the less part. Let 
P be the point in the line AB, P Q and 
PR perpendiculars to the other sides, are 
known; i?(7and AG are both known. 
Now, through the given point P, it is 




128 SURVEYING. 

required to draw PD, so that the triangle BPD, shall be the nth 
part of ABC. That is 



Whence, BD- 



n 
BO. AG 




nPQ 

Case. 6. When the given point is without the triangle. 

Let ABC be the given triangle, and P any given point without 
it. It is required to run a line from P, to cut off a given portion of 
the triangle ABC, or (which is the same thing) to divide the tri- 
angle into two parts having the ratio of m to n. Let PG be the 
line required. 

As P is a given point, AP is a line 
given in distance and position ; therefore, 
the angle PAH is known. 

Solution. — Put the angle PAH=u, 
CAB—v; then PAG=(u+v). Also 
put AG=x, AIf=y, AP=a, and the 
area of the triangle AHGf=mc, mc being a known quantity. 

Now, (by Prob. III. Mens.) \xy sin. v=mc (1) 

Also " " \ax sin. (w-f-v)=area APG 

And " " \ay sin. w=area PAH 

Therefore, \ax sin. (u-\-v) — \ay sin. u=mc (2) 

Or, % sin. (u+vj—y sin. ^=^f (3) 

From (1), we find y= which value substituted in (3) gives 

x ' x sin. v 

. t , x 2 mc sin. w 2mc 

x sin. (w-r-v) — - = 

x sm. v a 

__, „ . , , x 2 mc 2mc sin. u 

Whence, x 2 sin. (u-f v) x = — = 

a sm. v 

- _ 9,mc 2mcsin. u 

Or, s 2 — — . . , . *=- , — — — 

a sm. («H-v) sm. v sm. (tt-t-v) 



n« f __ mC 1 / ?K 2 c a 4. 2mcsin.tt 

ineretore,s_ a gin ( w+v )=i=V a2 sin.2( w -f- v ) sin.v sin.(w-|-t;) 



DIVISION OF LANDS 129 

EXAMPLES. 

1. In the triangle ABC, the side AB=23.645 chains, AC=11.51 
chains, and £0=12.575 chains. 

The given point P from the angle A, is distant 10 chains, at an 
angle of 40° from the line AC. 

It is required to draw a line from this given point P, through the 
triangle, so as to divide it into two equal parts. 

Whereabouts on AB will P O intersect ? 

The angle BAC=M° 17' 19"=v. PAH=40°=u. Therefore 
PAG=7\° 17' 10"=(«-H>.) 

The area of the triangle ABC is 107.52 square chains. The 
part to be cut off by the triangle AUG is therefore =53.76=m. 

We must use the natural sines, or the logarithmic sines if we omit 
them in the index. 

mc log.- - - 1.730464 

asin.(w+v)log. - - - 0.976406 

mc =5.676 log. 0.754058 

a sin. (u-\-v) 

2 



32.22 log. 1.508116 



a 2 sin. 2 (tt-H0 
2mc log. - - - 2.031494 

sin.w log. - - — 1.808067 

2wicsin. u - 1.839561 

sm. v sm.(u-\-v) „ - - — 1.691866 

Zmcsm.u =14051 j 2.147695 
sin. v sin. (u-\-v) 

The part of the formula under the radical is therefore (32.22-f* 
140.51) or 172.73. 



Whence ^=5.676^7172.73=18.819, or —7.467. 

Remabk. — The minus sign means opposite in direction, and if we pro- 
duce AC and AG (see last figure,) to the left of A, a line drawn from P 
through a point which is (7.467) to the left of A, will form a triangle below 
the line AG, which will be equal to AHG. 
9 



130 SURVEYING 

2. We have a right angled triangle whose base is 47.87 chains, and 
perpendicular 54.46 chains. From a given point without it, we are 
required to run the center of a straight road, to leave one third of the 
triangle on one side and two thirds on the other. 

From the acute angle at the base, the distance to the given point is 
20 chains, and the line to it makes an angle with the hypotenuse of 30°. 

Remark. — If P is a given point, its distance 
and direction from one of the angular points 
must be given ; and if the distance and direction 
from one of the angular points is given, the dis- 
tances and directions from all of them are virtually- 
given ; thus, if we have AP, AC, and the angle 
PAC, we have CP, and the angle ACP, and we 
may theorize on the triangles PCH, CHG' as well 
as on APH and AHG. 

The area of the triangle AB (7=1304.94 square chains. 
One third of this is mc= 434.97. 

BA (7=2/=49° 12' 20". PA C=u=30°. (u+v)='79 12' 20' . 
AP=a=20. AH=y. AG=x. 




1.293268 



mc log. - 2.638459 

a=20 log. - 1.301030) 
sin. (u-\-v) log. — 1. 992238 j 

. mc , =22.142 log. 1.345191 

asin.(wi-v) 

2 



=490.34 log. 2.690382 



a 2 sm. 2 (u-{-v) 

2mc log. 2.939489 

sm.u - — 1.698970 



2mc sin. u - - - - 2.638459 
sin. v sm.(u+v) - - - — 1. 871339 

=584.98 log. 2.767120 



sin. v sin. (u-\-v) 



Whence, #=22.142± A /1075.32=54.932 r —10.648. 

Here ^4^=54.932. AB=47.S7. Hence 5(7=7.062. Having 
AP, AG, and the angle PAG, we can compute the angle APG. 



DIVISION OF LANDS. 



131 




PROBLEM II. 

To divide a triangle into three parts having the ratio of the three 
numbers m, n, p. 

Case 1. Bylines drawn from one angle of the triangle to the 
opposite side. 

Let ADE be the triangle and A the 
angle from which the lines are to be 
drawn. 

Divide DE the opposite side into 
parts in the ratio of m, n, and p, and 
from the points of division C and B, 
draw A C, AB, and the triangle is divi- 
ded as required. 

Demonstration. — The areas of triangles are as their bases multiplied 
into their altitudes, but here all the triangles have the same altitude ; 
therefore multiplying the bases into that altitude gives the same 
proportional product, and the areas of the triangles are as m, n, p. 

Case 2. By lines parallel to one of the sides. 

Let ABC be the triangle. Divide its 
numerical area into three parts in the ratio 
of m, n, p. Conceive the problem solved 
and DF, EG, the division lines parallel 
to AB. We are to determine the numerical 
values of CD, and CE. CB is known, 
put it equal to a. Put CD=x. CE=y. 

Now as similar triangles are to one 
another as the squares of their homologous sides, therefore 
x 2 : a 2 : : m : m-\-n-\-p. 

Whence, #=«*/ . 

In the same manner, 




m 



m-\-n-\-p 
/ m-\-n 



m-\-n-\-p 

In this manner we might divide the triangle into any proposed 
number of parts, having given ratios. 

Case 3. By lines drawn from a given point on one of the sides of 



the triangle. 

o 



132 SURVEYING 

Let AJBCbe the given triangle, and P 
the given point on the side AB. 

It is required to draw lines from P, as 
PD and PE, dividing the triangle into 
three parts mc, nc,pc, that is, assume the 
given numerical area to be (mc-\-nc-\-pc), 
then the required parts will be mc, nc, and^?c* Put AD=x, then 
(by Prob. Ill, Mens.) 

| ax sin. A=mc or x=. * mc 





a sin. A 

By comparison, ?/— P c 

b sin. B 

When wic and pc are cut off, nc is left. Having a and x, the angle 

^LDP is easily determined. 

In a similar manner we can divide the triangle into any proposed 
number of parts, by lines drawn from the given point P. 
Case 4. By lines drawn from a given point within the triangle. 

Let ABC be the given triangle, and P 
the given point within it. 

A variety of lines may be drawn from 
P, to divide the triangle into the parts 
required. Conceive PD, PF y and PE 
to make the requisite division. 

As P is a given point, AP, PB, and PC are known lines, and 
the angles DAP, PAE are known angles. 

Take ADs=.b y any convenient assumed value. Take AE=x. 
Put AP=a. Then, 

\ab sin DAP= area A DAP 

Also, \ax sin PAE= area A PAE 

* Suppose we had a triangular piece of ground containing 320 square rods, 
and we wished to divide it into three parts in the ratio of 2, 3, and 5, what is 
the area of each of the parts ? 

We decide it thus : wi=2. w=3. p=5. c is at present unknown, but, 
mc-{-ne-}-/)c=320 

That is, 10c=320 or c=32. 

Whence, mc=64. nc=96. j?e=160. 

The quantity c becomes known on dividing the area, and the part* separately 
mc, nc, and pc, are always knoton. 



DIVISION OF LANDS 



133 



Conceive the triangle ABC, divided into three parts, in the ratio 
of m, n,p ; and conceive ADPE to be one of these parts repre- 
sented by mc. Then 

ab sin. DAP-\-ax sin. PAE—2mc 

2mc — ab sin. DAP 

Whence, x= = — p , ^ 

a sin. PAE 

Had we taken b greater than we did, x would have been less, and 
a variety of lines could be drawn as well as PD and PE, and the 
same area cut off. 

Having x, we have EB as a known quantity, and by the two 
triangles, PEB and BPF, we determine y in precisely the manner 
as we found x : thus, we have two parts of the triangle mc and pc, 
and, consequently, the remainder DPEO corresponds to nc. 

Case o. By lines drawn from a given point without the triangle. 
Let ABC be the triangle, and 
P the given point without it. 

Divide the numerical area of 
the triangle into the three re- 
quired parts, mc, nc, and pc, as 
in former cases. Draw PD, 
cutting off the portion mc, as 
in Case 6 of the last problem; 
then cut off the two portions ( mc-\-nc) by the line PG: and the 
portion pc will be left. Or, we may cut off pc, and the portion nc 
will be left. 

PROBLEM III. 

To divide a triangle info three parts, having the ratio of m, n, and 
p, by three lines drawn from the three angular points to some point 
within. 

Divide any side, as A C, into three 
parts in the proportion of m, n, 
and p. 

Let Aa represent the portion cor- 
responding to m, and Cc the part cor- 
responding tojo. 

Through a, draw ab parallel to AB, 





134 SURVEYING. 

and through c, draw cd parallel to CB. Where these two lines in- 
tersect is P, and the triangle ABC is divided into three triangles, 
APB, CPB,amlAPC. 

Demonstration. — Any triangle having AB for its base, and its 
vertex in the line ab, will have the same ratio to the triangle ABC, 
as Aa has to A C, that is, as m to m-\-n-\-p. 

Also, the triangle CPB is to ABC, as Cc is to CA, that is, as^> to 
m-\-n-\-p, for triangles on the same base are to one another as their 
altitudes. If these two triangles, APB and CPB, are in due pro- 
portion, the third one, AP C, is in due proportion, of course. 

PROBLEM IV . 

To divide a quadrilateral into two parts, having any given ratio 
m to n. 

Case 1. By a line drawn from a given point in the perimeter. 

Let A BCD be the given quadri- 
lateral, and P the given point in the 
side AB. 

It is required to determine the 
matrnitude, and the direction of the line PO, which divides the 
figure into parts in the ratio of m to n. All the sides and angles 
of the quadrilateral are known, and its area is known. AB and 
D C are, or are not parallel ; if they are parallel the figure is a tra- 
pezoid, then the method of finding O, in the opposite side, is 
easy and obvious. If AB and CD are not parallel, we can pro- 
duce them and form a triangle in the one direction or the other ; by 
this figure we form the triangle BCE, whose area we may repre- 
sent by t. 

As B C, and the angles as B and C are all known, the triangle 
EB C is determined in all respects. 

As PB is known, PE is known, and designate EG by x. Let 
cm and en designate the portions of the quadrilateral after it is di- 
vided, and let cm represent the part BPGC. 

Put PE=a. 

Now ( by Problem III, Mensuration ), we have 
\ax sin. E=t-\- cm. 




DIVISION OF LANDS. 135 

2t+2cm 
Whence, x =c^^TE 

We now have the numerical value of x, from which we subtract 
EC, and we have CG, which being measured from C will give the 
point G, through which to draw the line from P, to divide the 
figure as required. 

Case 2. By a line making a given angle with one of the sides. 

If the division line makes a given angle with one side, it must 
also make a known angle with the opposite side. 

Taking the last figure, conceiving PG to take a given direction 
across AB and CD, so as to cut off the area mc. 

As in the former case, let / represent the area of the triangle EBC, 
to this add mc, and we have the area of the triangle P GE. But in 
this case P is not a given point, and EP is not known. 

Put EG=x. Let P represent the given angle at P, and G the 

given angle at G. Now, by trigonometry, 

sin. P :x : : sin. G : EP 

~ „ D sin. G 

Or, EP=- -x 

sin. P 

(Prob. III. Mens.) Sin.^simg 2 ^ 

v J 2sm.P r 



Whence, x= ./ ' (n+2mcfl 



sin. P 



sin. G Sin. E 

From # we take EC, measure off the remainder along CD, to the 
point G, there making the given angle, and the figure will be di- 
vided as required. 

Case 3. By a line drawn through a given point within the quad- 
rilateral. 

Let ABCD be the quadrilate- 
ral as before, and P the given 
point within it ; and as P is the 
given point, EP is a known line, 
and the angles PER, PEG are known. 

Let t equal the area of the triangle EBC, as before, and mc the 
area GECB. Put EH=x, and EG=y. 





136 SURVEYING. 

Now we have a problem precisely like Case 4, Problem I, of this 
chapter ; therefore, further explanations would be superfluous. 

Case 4. By a line drawn through a given point without the 
quadrilateral. 

Let ABCBhe the quad- 
rilateral as before, and P the 
given point without it. 

By producing the two 
sides AB CD, we form the 
triangle ABE. Let the area of the triangle B CE be represented 
by t, and the part GH CB by mc, then from a given point P, with- 
out a triangle, we are required to draw a line PIT, to divide the 
triangle into two given parts, and this is Case 6, of Problem I of this 
chapter, which has been fully investigated. 

Remark. — By extending the principles of these several cases we may divide 
a quadrilateral into three or more parts. 

PROBLEM V. 

To divide any polygon (regular or irregular) into two parts having 
a given ratio, m to n, by a line drawn through a given point. 

Case 1. When the given point is on one of the sides of the polygon. 

Let ABODEF be the polygon, and 
(mc-\-nc) express its numerical area. Let P 
be the given point on the side AB. Let the 
surveyor run a random line as near to the 
line required as his judgment permits, and 
generally it will be best to run a line from 
P to one of the opposite angular points. 

In this figure, let PE represent such a random line, and let the 
surveyor compute the area of the figure PAFE, thus cut off, which 
area will be equal to, or greater, or less than one of the required 
parts. We will suppose it less ; then subtract it from the required 
portion mc, and let the triangle PEG represent that known difference, 
which we shall designate by t. 

PE is known, the angle PEG is known ; and put EG~-x 




Then, 


iPExsm.PEG=t 


Whence, 


2t 
PEsin. PEG 



DIVISION OF LANDS. 



137 



This determines the point G, and PG divides the polygon as 
required. 

Case 2. When the given point is within the polygon. 

Let AB GDEF be the polygon as before, and (mc-\-nc) express 
its numerical area ; also, let P be the given point within. Through 
P let the surveyor run the random 
line, HPK, measuring from Hto P, 
and from P to K, let him also 
observe the angles that this line 
makes with the sides of the polygon 
AFund CD, and compute the area 
HABCK, and note the difference 
between it and mc, the required 
portion of the polygon ; call this 
difference d, a known quantity. 

Let hPk represent the true line through P, which divides the 
polygon as required ; but this line diminishes the area HABCK, by 
the triangle PKk, and increases it by the triangle PUh. 

Therefore the difference of these triangles must equal d. 

The sine of the angle AHP, has the same numerical value as the 
sine of PHh, and the sine of the angle PKD has the same numeri- 
cal value as the sine of the angle PKk. 

Put the angle AITP=u, and the angle PKD—v; let the acute 
verticle angles at P be designated by the letter P. 

Let HP=a, PK=b, hP=x, Pk=y 




In the triangle Phil, we have 



sin. u : x : 



Whence, 



sin. (u — P) 
a sin. u 



Similarly, 



sin. (u — P) 
b sin. v 



y- 



(i) 

(2) 



sin. {y — P) 
The area of the triangle PhB=^ax sin. P 
Also, PkK=\by sin. P 

Whence, b sin. Py — a sin. Px=2d (3) 

By substituting the values of x and y, taken from (1) and (2), 
we have, 



138 



SURVEYING. 



b 2 



P sin. v 



-a 2 



sin. P sin. 



■■2d 



(4) 



sin. (v—P) sin. (u — P) 

Equation (4) contains only one unknown quantity P, the value of 
P, or the angle EPh can therefore be deduced. 

sin. P sin. v \ n2 ( sin. P sin. ^ \ 

V \sin. w cos. jP— cos. w sin. />/ 



b 2 



(, 



.sin. v cos. P — cos. v sin. P, 
•=2d.' 

Dividing the numerator and denominator of the first fraction by 
(sin. P sin. v), and of the second fraction by (sin. P sin. u.), recol- 
lecting that cosine divided by sine gives cotangent. Thus we shall 

obtain ,.,, , .„, __ u (5) 



( \ y«'( » v 

\cot. P — cot. vj \cot. P — cot. uj 



This last equation shows the surveyor that if he can make it conven- 
ient to run his random line from P, perpendicular to one of the sides, 
his equation will be less complex. For instance, if AIIP=90°, its 
cotangent will be 0, and cot. u would then =0, and equation (5) 
would become. 

b 2 a 2 



cot. P — cot. # cot. P 

For the sake of convenience put cot. P=z, and cot. v=c. 

Then 

b 2 a 2 



(6) 



Or, 



z — c 

,2 



=2d 



j_/a 2 —b 2 \ a 2 ( 
+ \-^d— C ) Z =2d 



2d " J"~~ Zd 
The numerical value of z will be the numerical value of cot. P ; its 
logarithm taken, and 10 added to the index will be logarithmic cot. 
in our table. The same remarks will apply to cot. v or c. 

Case 3. When the given point is without the 
polygon. 

Let ABODE be the polygon, and P the 
given point without it. 

From the last case we learn that the surveyor 
had better run his random line perpendicular 
to one of the sides, therefore let PHG be the 
random line, perpendicular to AB. 




DIVISION OF LANDS. 



139 



As before, compute the area, AEGH, subtract it from mc, the dif- 
ference is the difference between the triangles PGL and PHK. 
Draw PKL the line that divides the polygon as required. 

Put PH=a, PG=b, PK=x, PL=y, angle 11=90°, angle PGE 
c=w, and the angle at P, designated by P. 

The triangle PHK=\ax sin. P 
PGL=\by sin. P 
Whence, b sin. Py — a sin. P x=2d. (1) 

Here (2) represents a similiar quantity as in the last case. 
In the triangle PHK, we hare 



1 : x : : cos. P a, or x- 



cos. P 



In PGL, sin. u : y : : sin. (w — P) : 5. 

b sin. w 



y= 



sin.(w— P) 

When the values of x and y are substituted in ( 1 ) we have 
sin. P sin. u sin. P 



b 2 



Or, 6 2 



Or, 



sin. (u — Pj 

(sin. P sin. 
sin. u cos. 



cos. 



P — cos. m sin. P 
b 2 a 2 



)-» 



(2) 
(3) 
(4) 



o sin. P 
2 ^TP ==2d (5) 



:=2d 



cot. P— cot.w cot. P"~ ~* (6) 

This equation is exactly similar to equation (6) of the last case, 
and it is reduced in the same manner. 



PROBLEM VI. 

To divide a polygon into three or more parts, having a given ratio, 
m, n, p, q, by lines passing through a given point. 

This problem admits of three cases. 

Case 1 . When the given point is 
on one side of the polygon. 

Divide the numerical area of the 
whole into parts, mc, nc, pc, qc, cor- 
responding to the given ratio. Unite 
these into two parts (?nc-\-nc) and 
(pc+qc). 




140 



SURVEYING. 




From the given point P, draw PG, by Case 1, Problem V, 
so as to divide the polygon into the two parts (mc-\-nc) and [pc-\-qc). 

We have now to divide the polygon PAEQ into two parts, mc, 
nc, by the line PL, and the polygon PBCDG into two parts, pc 
and qc, by the line PH. 

Case 2. When the given point is within the polygon. 

Let ABODE be the given poly- 
gon and P the given point within. 

Draw hh through P, by Case 2, 
Problem V, so that the area AhhDE 
shall equal mc, and the area hhOB 
shall equal (nc-\-pc), when the whole 
is required to be divided into three 
parts in the ratio of m, n, p. 

When the whole is to be divided into four parts, in the ratio of m, 
n, p, q, then draw hJc, so that one portion shall be (mc+nc) and 
the other (pc-\-qc). 

Then we have P as a given point, in one side of the polygon, 
AlikDE, to divide it into two parts, in the ratio m to n, and P a given 
point on one side of the polygon, KkCB, to divide it into two parts, 
in the ratio of p to q, and this is done by Case 1, Problem V. 

Case 3. When the given point is without the polygon. 

Let ABCDEF be the given poly- 
gon, and P the given point without it. 

Divide the numeral area into the 
required proportional parts, mc, nc,pc, 
<fcc, as many as required. 

From the point P draw the line PH, 
as directed in Case 3, Problem V, di- 
viding the polygon into two parts, mc 
and (nc-\-pc-\-<kc). 

Then divide the polygon, GHED OB, into two parts, one of which 
is nc, and the other (pc-Jrqc, &c), and thus we can proceed and 
cut off one portion after another, as many as may be required. 

The application of the foregoing principles will meet any case that 




DIVISION OF LANDS. 141 

can occur in the division of lands ; and we now close this subject 
with the following practical 

EXAMPLES . 

1. A triangular field, whose sides are 20, 18, and 16 chains, is to 
have a piece of 4 acres in content fenced off from it, by a right line 
drawn from the most obtuse angle to the opposite side. Required the 
length of the dividing line, and its distance from either extremity of the 
line on which it falls ? 

Ans. Length of the dividing line, 13 chains, 89 links, if run 
nearest the side 16. Distance it strikes the base from the next most 
obtuse angle is 5.85 chains. 

2. The three sides of a triangle are 5, 12, and 13. If two-thirds 
of this triangle be cut off by a line drawn parallel to the longest side, it 
is required to find the length of the dividing line, and the distance of 
its two extremities from the extremities of the longest side. 

Ans. Distance from the extremity on 5, is 5(^/3 — ^/2); on the 
side of 12, it is 12(^/3—^/2) ; both divided by ^"3. 
The division line is 13,/f. 

3. It is required to find the length and position of the shortest possible 
line, which shall divide, into two equal parts, a triangle whose sides are 
25, 24, and 7 respectively. 

Remark. — It is obvious that the division line must cut the sides 25 and 24, 
and to make it the shortest line possible, the triangle cut off must be Isosceles. 

Ans. The division line makes an angle with the sides 25 and 24 
of 81° 52' 1 1", and its length is 4.899. 

4. The sides of a triangle are 6, 8, and 10. It is required to cut 
off nine-sixteenths of it, by a line that shall pass through the center of 
its inscribed circle. 

Ans. The division line cuts the side of 10, at the distance of 7.5 
from the most acute angle, and on the side of 8, at the distance 6 
from the most acute angle. 

5. Two sides of a triangle, which include an angle of 70°, are 14 
and 1 7 respectively. It is required to divide it into three equal parts, 
by lines drawn parallel to its longest side. 



142 SURVEYING. 

Ans. The first division line on the side 17, cuts that side at the 

17 ... .17/2 

distance— -; the second division line — ^-. The side 14 is cut at 

V 3 V 3 

iiand l*& 

V 3 V 3 

6. Three sides of a triangle are 1751, 1257.5, and 2364.5. The 
most acute angle is 31° 17' 19". This triangle is to be divided into 
three equal parts by lines drawn from the angular points to some point 
within. Required the lengths of these lines. 

Ans. The line from the most acute angle is 1322.42, and from the 
next most acute angle 1119 

7. The legs of a right-angled triangle are 28 and 45. Required the 
lengths of lines drawn from the middle of the hypotenuse, to divide it 
into four equal parts. 

Ans. A line drawn from the middle of the hypotenuse to the 
right angle, divides the triangle into two equal parts. 

8. In the last example, suppose the given point on the hypotenuse at 
the distance of 13 from the most acute angle, whereabouts on the other 
sides will the division lines fall to divide the triangle into three equal 
parts f 

N". B. The sine of an acute angle to any right-angled triangle is 
equal to the side opposite that angle divided by the hypotenuse. 

Ans. Both division lines fall on the side 28, distance of the first 
from the acute angle 12Ji, of the second 24|| 

9. There is a farm containing 64 acres, commencing at its south 
westerly corner, the first course is North 15° E., distance 12 chains ; 
the second is JV. 80° E. (distance lost), the third S. (distance lost), 
the fourth is JV. 82° W. (distance lost), to the place of beginning. It 
is required to determine the distances lost. 

Observation. — Extend the northern and southern boundary westward, and 
thus form a triangle on the west side of 12. 

Ans. The 2nd side is 35.816 ch. 3rd, 23.21 ch. 4th, 38.76 ch. 

The two following problems are from Gummere's Surveying, and 
are considered very difficult. 




DIVISION OF LANDS. 143 

1. There is apiece of land hounded as follows : 
Beginning at the south-west corner ; thence, 

1. N. 14° 00' W, Distance 15.20 chains=a; 

2. K 70° 30' E., " 20.43 " ==b ; 

3. S. 6°00'E., " 22.79 " =c; 

4. 1ST. 86° 30' W., " 18.00 " =d. 

Within this lot there is a spring ; the course to it from the second 
corner is S. 75° E., distance 7.90 chains. It is required to cut off ten 
acres from the west side of this lot, by a line running through the spring. 
Where will this line meet the fourth side, that is, how far from the 
first corner ? Ans. 4.6357 chains. 

First make a plot of the field. It is as here represented. 

Produce the sides b and d, the second 
and fourth, until they meet at G. Let S 
be the position of the spring, and join SO. 
We may or may not find the contents of 
the field* : it is not necessary for the 
location of the fine LSR. 

It is necessary to find the area of the triangle AB G. Conceive 
a meridian line run through B ; then we perceive that the angle 
ABG=U°+70° 30'=84° 30'. Conceive also a meridian line to 
be run through A, and then we perceive that the angle BAG— 86° 
30'— 14°=72° 30' ; whence A GB=23°. With the angles and the 
side ^£=15.20, we readily find AG=3$.72, BG=37.10, and the 
area ^4£i?=280.65 square chains. 

It is necessary to find the line GS and the angle BGS. From 
the given direction of the lines BC and BS, we find the angle 
6^£=145° 30' ; and then from the triangle GBS, we find 
BGS=5° 51' 30", and GS= 43.83. Also we have the angle 
SGA=\7° 8' 30' . 

To the area of the triangle ^£5=280.65, add 10 acres, or 100 
square chains : then the area of the triangle GLITmust equal 380.65 

* When we have four sides only, and all the angles, as in this field, the best 
method of finding the contents is by conceiving it to be two triangles. Thus in 
this case the area is represented by 

I ab sin. ABC-\-\ dc sin. CD A. 



144 SURVEYING. 

square chains ; but GL and GH are both unknown. Put GL=y, 
GH—x : then we shall have the equation. 

xy sin. 23°=2(380.65). (1) 

It is obvious that the sum of the two triangles ZGS, SGH is 
equal to the triangle GLH. 

But ##=m=43.83, sin. 23°= P, sin. (17° 8' 30")=§, sin. 
(5° 51' 30")=i2, and 2(380.65) or 761.3=a : then we have 

Pxy=a, ( I ) 

and Rmy-\- Qmx=a, (2) 

From (1), y— — . This value put in (2), gives 

Px 

»J^ +Qmx=a , (3 ) 

whence, *-JLf = -||. (4) 

n ft 7? 

We now find the numerical values of and by logarithms, 

mQ PQ 

as follows : 

As our radius is unity, we diminish the indices of the logarithmic 
sines by 10. 

log. a, 2.881556 log. a, 2.881556 

log. m, 1.641771 log. .#,—1.008880 

log. §,—1.469437 

1.890436 



1.111208 1.111208 



58.932 1 .770348 log. P,— 1 .59 1 878 
log. §,—1.469437 



—1.061315 —1.061315 



674.72 2.829121 

Equation (4) now becomes 

#2_58.932z=— 674.7; 
whence x— ##=43.366 : from which subtract (£4=38.72, and 
we have AH the distance required =4.646, which differs from the 
given answer about one link of the chain. 




DIVISION OF LANDS. 145 

Lemma. Find the point in any trapezoid, through which any 
straight line which meets the parallel sides will divide the trapezoid into 
two equal parts. 

Let AB CD be the trapezoid. 

Bisect the parallel sides AB and CD in 
the points n and m. Join mn, and bisect 
mn in 0, and is the point required. 

Any line meeting the parallel sides, and passing through 0, will 
divide the trapezoid into two equal trapezoids. It is obvious that 
the line mn divides the figure into two equal parts, because the sum 
of the parallel sides is the same in each. Now draw any other line 
through 0, as p Oq : the trapezoid pqBD—mnBD ; because the 
triangle Omp=qOn, and one triangle is cut off and the other is put 
on at the same time. The triangles are equal, because mO=.On, 
the angle pm 0= Onq, and the opposite angles at are equal : 
therefore pm—nq ; and whatever more than Cm is taken on one 
side, the equal quantity qn less than An is taken on the other side. 

Another method of finding the point 0, is to bisect A C in H, and 
draw HO parallel to AB or CD, and equal to one-fourth the sum 
of^LSand CD. 

2. There is apiece of land bounded as follows : 
Beginning at the westernmost point of the field ; thence, 

1. JV. 35° 15' E., 23.00 chains ; 

2. N. 75° 30' E., 30.50 « 

3. S. 3° 15' E., 46.49 " 

4. N. 66° 15' W., 49.64 " 

It is required to divide this field into four equal parts, by two lines, 
one running parallel to the third side, the other cutting the first and 
third sides. Find the distance of the parallel line from the first corner 
measured on the fourth side, and the bearing of the other line. 

Ans. Distance to the parallel, 32.50 chains ; Bearing of the other 
side, S. 88° 22' E. 
10 



146 SURVEYING. 

CHAPTER VII. 

TRIANGULAR SURVEYING: THE PLANE 

TABLE— ITS DESCRIPTION AND USES: 

MAPPING: MARINE SURVEYING. 

Triangular Surveying, as here understood, requires the actual 
measurement of only one line, and all other lines can be deduced 
from this by means of observed angles forming triangles, of which 
this measured line forms a base of the first triangle in the series or 
chain of triangles. 

o 

Some of the other lines however should be measured after being 
computed, as a test to the accuracy or inaccuracy of the operations. 

Let AB represent a base line 
which must be very accurately 
measured, for any error on AB will 
cause a proportional error in every 
other line. 

If at A we measure the angles 
BA C, BAD, and at B we measure 
or observe the angles ABC, ABB, 
we then have sufficient data to deter- 
mine the points C and D, and the line CD. 

With equal facility that we determine the point C, we can deter- 
mine the point E, or F, or G, or any other visible point. 

Thus we may determine all the sides and angles of the figure 
CEFGHD, or any visible part of it, by triangulating from the 
base AB. 

The lines forming the triangles are not drawn, except those to the 
points C and D ; we omitted to draw others to avoid confusion. 

After any line, as FG, has been computed, it is well to measure it, 
and if the measurement corresponds with computation, or nearly so, 
we may have full confidence of the accuracy of the work as far as it 
has been carried. 

We may take CD as the base, and determine any visible number 
of points, as A, B, H, F, G, .&c, trace any figure and determine its 
area, or show the relative positions and distances of objects from 
each other, such as buildings, monuments, trees, &c. 




THE PLANE TABLE. 147 

But to make the computation, triangle after triangle, for the sake 
of making a map, would be very tedious, and to measure every side 
and angle would be as tedious, and to facilitate this kind of operation 
we may have an instrument called the 

PLANE TABLE. 

The plane table is exactly what the name indicates ; it is a plane 
board table, about two feet long, and twenty inches wide, resting on 
a tripod, to which it is firmly screwed, yet capable of an easy motion 
on its center, having a ball and socket like a compass staff. 

Directly under the table is a brass plate, in which four milled 
screws are worked, for the purpose of adjusting the table, the 
screws pressing against the table. 

To level the table, a small detached spirit level may be used. 
The level being placed on the table over two of the screws, the 
screws are turned contrary ways, until the level is horizontal ; after 
which it is placed over the other two screws, and made horizontal in 
the same manner. 

The table has a clamp screw, to hold it firmly during observa- 
tions, and also a tangent screw, to turn it minutely and gently, after 
the manner of the theodolite. 

The upper side of the table is bordered by four brass plates, 
about an inch wide, and the center of the table is marked by a pin. 

About this center, and tangent to the corners of the table, conceive 
a circle to be described. Suppose the circumference of this circle to 
be divided into degrees and parts of a degree, and radii to be drawn 
through the center, and each point of division. 

The points in which these radii intersect the outer edge of the 
brass border, are marked by lines on the brass plates ; these lines 
of course show degrees and parts of degrees ; they are marked from 
right to left, from to 180° on both sides, but on some tables the 
numbers run all the way round, from to 360°. 

Near the two ends of the table are two grooves, into which are 
fitted brass plates, which are drawn down into their places by screws 
coming up from the under side. The object of these grooves and 
corresponding plates, is to hold down paper firmly and closely to 
the table. 



148 SURVEYING. 

The paper before being put on, should be moistened to expand it, 
then carefully drawn over the table, and fastened down by the plates 
that fit into the grooves ; on drying, it will fit closely to the table. 

A delicate fine edged ruler is used with the plane table, it has 
vertical sights ; the hairs of the sights are in the same vertical plane 
as the edge of the ruler. 

A compass is sometimes attached to the table, to show the bear- 
ings of the fines ; but the most practical mathematicians prefer each 
instrument by itself. 

The plane table may be used to advantage for three distinct 
objects. 

1. Forth© measurement of horizontal angles. 

2. For the determination of the shorter lines of a survey, both as 
to extent and position. 

3. For the purpose of mapping down localities, harbors,, water- 
courses, &c. 

1 . To measure a horizontal angle. 

Place the center of the table over the angular point, by means 
of a plumb. Level the table, then place the fine edge of the ruler 
against the small pin at the center ; direct the sights to one object, 
and note the degree on the brass plate ; then turn the ruler to the 
other object, and note the degree as before. 

The difference of the degrees thus noted, is the angle sought. 

If the ruler passed over 0, in turning from one object to the other, 
subtract the larger angle from 180°, and to the remainder add the 
smaller angle, for the angle sought. 

2. To determine lines in extent and position. 

Let CD be a base line; having 
the paper on the table, all dried and 
ready for use. Place the table over 
C, so that the point on the table, 
where we wish C to be represented, 
shall fall directly over 0; and place 
the position of the table so that CD 
shall take the desired direction on 
the table. 




THE PLANE TABLE. 149 

Now level the instrument, and clamp it fast ; it is then ready for 
use. 

Sight to the other end of the base line, and mark it along the fine 
edge of the ruler. 

In the same manner sight along the direction of CE, and mark 
that direction in a fine lead line, that can be easily rubbed out, the 
point E is somewhere in that line. 

Sight in the direction of F, and mark the line on the paper ; F is 
somewhere in that line. In this manner sight to as many objects as 
desired, as G, H, B, A, &c. 

Now the base on the paper, may be as long, or as short as we 
please ; suppose the real base on the ground to be 1 200 feet ; this 
may be represented on the table by 3, 4, 5, 10, or 12 inches, more 
or less ; suppose we represent it by 6 inches, then one inch on the 
paper, will correspond with 200 feet on the ground (horizontally). 

Take CD six inches, and place a pin at D, remove the instrument 
to the other end of the base, and place D of the table right over 
the end of the base, by the aid of a plumb, and give the table such 
a position as will cause CD on the table, to correspond with the 
direction of the base. 

Level the table and clamp it. Now, if CD on the table, does 
not exactly correspond with the direction of the base on the ground, 
make it correspond by means of the tangent screw. 

Now from D, by means of the ruler and its sight vanes, draw 
lines on the paper, in the direction of the points E, F, G, If, B, A, <fec; 
and where these lines intersect those from the other end of the base, 
to the same points, is the real localities of those points, in proportion 
to the base line. Lines drawn from point to point, where these lines 
intersect, as EF, FO, GH, &c, will determine the distances from 
point to point, at the rate of 200 feet to the inch. 

Lines drawn from the center of the table, parallel to FE and FG, 
will determine the angle EFG, in case the angle is required. After 
the points E, F, G, &c, have been determined, the light pencil lines 
to them, from the ends of the base, may be rubbed out, except those 
that we may wish to retain. 

Here we perceive the utility of the plane table ; we have a multitude 
of results, as soon as the observations are made. 



150 



THE PLANE TABLE. 



The plane table will give us at once, the relative distances of 
buildings from the base, and from each other, and if we are careful 
and particular, we can obtain the magnitudes of the buildings, as is 
obvious by the adjoining figure. 




This is most useful to an officer or a spy, who wishes as exact 
knowledge of an enemy's locality as possible. Or from a distant 
place AB, we may examine and measure any objects whatever, on 
the other side of a river, or give a correct delineation of the river 
itself. 

The plane table can be made very useful to civil engineers, for 
mapping the localities through which a canal or railroad passes. 
Take, for example, a railroad line, ABCDE, represented in the 
next figure. The lines being all measured and marked in distances 
of 100 or 200 feet, the bases are all ready where the line is 
straight. 

Set the table as at A, and draw lines to all objects that you wish 
to appear on the map, both to the right and to the left, — then move 
the instrument to B, drawing lines to the same object from, a corres- 
ponding base on the paper, and also draw lines to other objects fur- 
ther in advance on the line that may be seen from another base. 

The intersections of the lines from the extremities of any base to 
to the same object will locate the object. 

When all the objects are thus located, both to the right and to 
the left, we pass on to new objects, taking care to keep at least two 
of the old objects in sight, to connect one new observation with those 
previously taken. We now commence a series of observations from 
anew base, which base must take its proper relative position on tke 



SURVEYING 



151 




152 SURVEYING. 

paper, and if the paper on the table is not large enough, it must be 
taken off and new paper put on, and two of the objects on the old 
paper must appear on the new, and then these two papers can be 
put together so that the objects which are on both papers will coin- 
cide, and then the two papers will be the same as one, and thus 
we may put any number of papers together and form as large a map 
as we please. 

If the different bases are not in the same direction, the objects 
which are on two different papers on being put together will show 
it, and several papers put together may make a very inconvenient 
figure ; but they must be put together and then a square sheet of 
tissue paper put over the whole, and the map taken off. From the 
tissue paper the map can be put on any other paper. 

The engineers of Napoleon's army frequently made maps of the 
localities they were about to pass ; indeed it is a military principle 
never to go into an unknown locality, except in cases of absolute 
necessity. 

This subject naturally leads us to 

MARINE SURVEYING. • 

Marine Surveying is too extensive a subject to be fully investi- 
gated in any work like this. We shall only explain how to find 
shoals, rocks, and turning points in a channel, by ranging to objects 
on shore. 

In trigonometrical surveying, on shore, the observer is supposed 
to take his angles from the extremities of a base line, but in trigono- 
metrical surveying on water, the observer can take his angles only 
from single points which may be connected together by distant base 
lines on the shore. 

Important points along the shore are determined By taking lati- 
tude and longitude, and intermediate places, by regular land sur- 
veying. 

The localities of rocks and shoals are also determined by astron- 
omical observations, establishing latitude and longitude, in case no 
tend is in sight, or they are far from the shore ; but in the vicinity 
of the land, the determination of a point is commonly effected by the 
three point problem. 



MARINE SURVEYING. 153 

The three point problem is the determination of any point from 
observations taken at that point, on three other distant points where 
the distances of these three points from each other are known. 

It is immaterial how those points are situated, provided the three 
points and the observer are not in the same right line, the middle 
one may be nearest or most remote from the observer, or two of 
them may be in one right line with the observer, or all three may be 
in one right line, provided the observer be not in that line. The 
following example will illustrate the principle. 

Coming from sea, at the point D, I observed two headlands, A 
and B, and inland C, a steeple, which appeared between the head- 
lands. I found, from a map, that the headlands were 5.35 miles from 
each other ; that the distance from A to the steeple was 2.8 miles, 
and from B to the steeple 3.47 miles ; and I found with a sextant, 
that the ande ADC was 12° 15', and the an ale BBC 15° 30'. 
Required my distance from each of the headlands, and from the 
steeple. 

If the direction of AB is known, the direction of A C is equally 
well known. 

The case in which the three objects, A, C, 
and B, are in one right line may require illus- 
tration. 

At the point A, make the angle BAE= the 
observed angle CBB, and at B, make the angle 
ABE= the observed angle ABC. 

Describe a circle about the triangle ABE, 
join E and C, and produce that line to the circumference in D, 
which is the point of observation. Join AD, BD. The angle 
ADB is the sum of the observed angles, and AEB added to it 
must make 180°. 

The Trigonometrical Analysis. — In the triangle ABE, we have 
the side AB and all the angles, AE and EB can therefore be com- 
puted. 

In the triangle AEC, we now have A C, AE, and the angle CAE, 
from which we can compute A CE, then we know A CD. 

Now in the triangle A CD, we have A C and all the angles, whence 
we can find AD and CD. 




154 SURVEYING. 

When the bearing of the base line on shore is known, as it gen- 
erally is, and the bearings to its extremities, or even to one extrem- 
ity, are taken, the triangle is known at once. 

A pilot guides a vessel in and out of a port by ranging lines on 
the shore, minutely or approximately, as the case may require. 

We will illustrate this by a figure. Let the deep shaded lines 
represent the shores, A a light house on a rocky promontory. B 
another prominent object on the opposite shore ; the position of the 
arrow indicates the direction north and south. 

The faint dotted lines represent the boundaries of shoal water, 
over which it would not be prudent, if even possible, to conduct 
a vessel. The line a, b, 
d, JE, the center of the 
ship channel. All the pi- 
lots know that a line from 
A to a, which is nearly 
east south east, runs safe 
to the open sea, after 
passing the shoal coast 
near n. 

Now suppose a pilot 
boards a ship coming in from sea, sufficiently far from the coast, he 
directs her sailing so as to bring the light house at A to bear west 
north-west. He then sails toward the light house until he finds the 
object B bearing clue north of him. He then knows that the ship 
must be near a, the mouth of the channel. 

He continues the same course, and knows when he is about half 
way from a to b by the two objects, C and D, appearing in the same 
line. When C and D become fairly open, and C nearly north, 
and B not quite north-east, he is then at the turning point b of the 
channel. His course is then north, a little to the west, until the 
ship is nearly in a line between the two objects A and B. From 
thence, west north-west takes the ship directly through the proper 
channel into the harbor. 




LEVELING 



155 



C HA PTEE VIII. 



LEVELING. 

Two or more points are said to be on a level, when they are 
equally distant from the center of the earth, or when they are equally 
distant from a tranquil fluid, situated immediately below them. A 
level surface on the earth, is nearly spherical, and is not a plane ; 
it is everywhere perpendicular to a plumb line. 

Any small portion of a true level surface, cannot be distinguished 
from a plane ; and, therefore, when observations are taken in respect 
to level, within short distances of each other, the spherical form of 
the earth is disregarded, and the level treated as a plane. But when 
any considerable portion of surface is taken into account, the curva- 
ture of the earth's surface must be considered. 

The apparent level, at any point on the earth, is a tangent plane, 
touching the earth at that point only, and the true level is below this, 
and the distance below, depends on the distance from the tangent 
point. 

Let T be any point on the surface of the 
earth, at right angles to the plumb line from 
this point is the plane or apparent level ATB; 
but the true level, or the surface of standing 
water, would be the curved surface GTH. 

The distance A G, depends on the distance 
AT, and the radius of the earth CG or C T. 

From G, draw GD, at right angles to A ; 
then the two triangles ATC, AGD, are equi- 
angular and similar, and give the proportion 
CT:TA::DG: GA. 

Practically. — TA is a very short distance compared to CT, for 
TA, the distance within which we can take observations, is never 
more than two or three miles, while the distance CT is near 4000 
miles ; therefore, CT is nearly equal to CA, consequently, DA is 
nearly equal to DG, so near that we shall call it equal. 

Observe that TD=DG; hence DG=\TA. 




156 SURVEYING. 

Now, in the preceding proportion, in the place of DG, put its 
equal -J TA, and we shall have, 

CT : TA : : \TA : GA. 

Whence, AU-JjjjJ Also,,«=^ (1) 

That is, The square of the distance, divided by the diameter of the 
earth, is the distance between the apparent and the tme level. 

We can arrive at the same result by the direct application of the 
36th proposition of the third book of Euclid, or by the application of 
theorem 18, third book of Robinson's Geometry. 

Because A is a point without a circle, and A ^touches the circle, 
we must have 

AGx(2CG+AG)=ZT 2 

But 2 CG, which is the diameter of the earth, cannot be essentially 
or appreciably increased, by the addition of AG, which is at most, 
but a few feet, therefore, AG within the parentheses, may be sup- 
pressed without making any appreciable error. Then divide by 2 CG. 

AT 2 
Whence, AG=-——, the same as before. 

2CG 

If we take one mile for the distance TA, the value of GA will 
be 7 Jj-s = 8.001 inches. 

By comparing equations (1) we perceive that, 
GA :ga:\ TA 2 : Ta 

That is, The corrections for apparent levels, are in proportion to the 
squares of the distances. 

The correction for one mile is 8.001 inches; what is it for 10 miles? 
Ans. It is x inches ; then we have the following proportion, 

8.001 : x : : l 2 : 10* x— 800.1 inches. 

We have seen above, that the correction for one mile or 80 chains 
distance, on an apparent level, is 8.001 inches, what is the correction 
for the distance of 20 chains ? 

Let #=the correction sought, and the solution is thus, 
8.001 : x : : (80) 2 : (20) 2 

: : 4 2 : l 2 x= 0.500 inches. 



LEVELING. 



157 



In this manner, the following table was computed. 

Table showing the differences in inches, between the true and apparent 
level, for distances between 1 and 100 chains. 



Ch's. 


In's. 


Ch's. 


In's. 


Ch's. 


In's. 


Ch's. 


In's. 


1 


.001 


26 


~T 845 


51 


"3^255 


76 


7.221 


2 


.005 


27 


.911 


52 


3.380 


77 


7.412 


3 


.011 


28 


.981 


53 


3.511 


78 


7.605 


4 


.020 


29 


1.051 


54 


3.645 


79 


7.802 


5 


.031 


30 


1.125 


55 


3.785 


80 


8.001 


6 


.045 


31 


1.201 


56 


3.925 


81 


8.202 


7 


.061 


32 


1.280 


57 


4.061 


82 


8.406 


8 


.080 


33 


1.360 


58 


4.205 


83 


8.612 


9 


.101 


34 


1.446 


59 


4.351 


84 


8.832 


10 


.125 


35 


1.531 


60 


4.500 


85 


9.042 


11 


.151 


36 


1.620 


61 


4.654 


86 


9.246 


12 


.180 


37 


1.711 


62 


4.805 


87 


9.462 


13 


.211 


38 


1.805 


63 


4.968 


88 


9.681 


14 


.245 


39 


1.901 


64 


5.120 


89 


9.902 


15 


.281 


40 


2.003 


65 


5.281 


90 


10.126 


16 


.320 


41 


2.101 


66 


5.443 


91 


10.351 


17 


.361 


42 


2.208 


67 


5.612 


92 


10.587 


18 


.405 


43 


2.311 


68 


5.787 


93 


10.812 


19 


.451 


44 


2.420 


69 


5.955 


94 


11.046 


20 


.500 


45 


2.531 


70 


6.125 


95 


11.233 


21 


.552 


46 


2.646 


71 


6.302 


96 


11.521 


22 


.605 


47 


2.761 


72 


6.480 


97 


11.763 


23 


.661 


48 


2.880 


73 


6.662 


98 


12.017 


24 


.720 


49 


3.004 


74 


6.846 


99 


12.246 


25 


.781 


50 


3.125 


75 


7.032 


100 


12.502 



This table is of little or no practical use, for levelers rarely take 
sight to a greater distance than 10 chains, and at that distance the 
correction is only one-eighth of an inch, and if they put the level 
midway between two stations, they annihilate the corrections 
altogether. 

Suppose a level to be placed at T, midway between A and B ; the 
instrument will show them to be on the same level, as so they really 
are, for they are at equal distances from the center of the earth ; but 
if the observations were taken in reference to A and a. the apparent 
level would not show equal distances from the center of the earth, 
and a correction must be applied, if the difference of distances is 
more than four or five chains. 



158 SURVEYING. 

To comprehend the whole subject, we must now describe the 
modern 

SPIRIT LEVEL. 




The figure before us represents this useful instrument, apart from 
its tripod. 

Its principal parts are the telescope AB, to which is attached the 
leveling tube CD; the telescope rests in a bed, which is supported 
by posts yy, called the y's ; EE is a firm bar, supporting the y's. 
In S is a socket, which receives the central pivot of the tripod 
(which is not here represented). 

When the instrument is put upon its tripod, the tube S can be 
clasped on the outside, and held firmly by a clamp screw, it can 
then be moved horizontally, as minutely and readily as desired, by 
means of a tangent screw. 

The tripod contains a pair of brass plates, to the lower one the 
legs of the tripod are firmly attached, the other plate moves in all 
directions on its center, and is worked by four serews ; these are 
called the leveling screws ; these plates are purposely made small as 
a greater surety against bending : the four leveling screws are 
placed at the four quadrant points of the circle, and, with the center, 
form diameters at right angles. 

The eye-glass of the telescope is at A, the object glass at B. The 
screw V runs out the tube which holds the object glass, to adjust it 
to different distances. 

The telescope is fastened into the y's, by the loops r r, which are 
fastened by the pins p p. The telescope can be reversed in the y's, 
by taking out the pins p p ; opening the loops r r ; taking up the 
tube, turning it round, and again placing it in the y's ; then A will 



LEVELING. 159 

take the position of B, and B of A. The necessity of this construc- 
tion will appear when we describe the adjustment. 

At n n are two small screws that are attached to a ring inside of 
the tube ; this ring holds a horizontal spider line ; the object of 
the screw is to elevate and depress that spider line. 

At q q ( only one q can be seen in the figure ), are two screws 
that work another ring, which holds a perpendicular spider line,, 
which can be moved to the right and left by the screws q q. The 
two spider lines show a perpendicular and horizontal cross at the 
focus of the telescope. 

Before using the instrument it must be adjusted. The adjust- 
ment consists : 

1st. In making the center of the eye-glass and the intersection of the 
spider 's lines coincide with the axis of the telescope, and this line is 
called the line of collimation. 

2nd. In making the axis of the attached level, CD, parallel to the 

line of collimation, in respect to elevation. 

3rd. In making the attached level lie exactly in the same direction as 
the line of collimation. 

To make the first adjustment, the telescope is made to revolve 
in the y y s. 

To make the second adjustment, there is a screw a, which serves 
to elevate and depress the end of the leveling tube at 0. 

To make the third adjustment, there is a side screw b, which 
drives the end of the tube D to the right and left, as the case may 
require. 

First Adjustment. — Plant the tripod, place the instrument upon 
it, and direct the telescope to some well defined and distant object. 

Draw out the eye-glass at A, until the spider's lines are distinctly 
seen, then run out the object glass by the screw V to its proper fo- 
cus, when the object and the spider's lines will be distinct. Now 
note the precise point covered by the horizontal spider's line. 
Having done this, revolve the telescope in the y's half round, when 
the attached level will be on the upper side. See if in this position 
the horizontal spider's line appears above or below the same object. 

If this line should appear exactly in the same point of the object 



160 SURVEYING. 

as before, this spider's line is already in adjustment, but if it ap- 
pears above or below, bring it half way to the same point by means 
of the screws n n, loosening the one and tightening the other. 

Carry back the telescope to its original position, and repeat 
the observation, and continue to repeat it until the telescope will 
revolve half round without causing the horizontal line to rise or 
fall. This will show that the horizontal line is a diameter of the cir- 
cle of revolution, and not a chord of it. Make the same adjustment 
in respect to the vertical hair, and the line of collimation is then 
adjusted. 

Second Adjustment. — That is, to make the tube CD horizontally 
parallel to the line of collimation. Place the instrument properly on 
its tripod, and bring the horizontal bar EE directly over two of the 
leveling: screws ; turn these screws until the bubble d rests in the 
center of the tube. 

Now CD is on a level, but we are not able to say that the line of 
sight through the telescope, that is the line of collimation, is on a 
level also. To test this, take out the pins p p, open the loops r r, 
and take out the telescope with its attached level, and turn it end 
for end, put it back in its bed, and put the loops over and pin them 
down. If the bubble now rests in the middle, no adjustment is re- 
quired; if not, the bubble will run to the elevated end. In that case 
the bubble must be brought half way back by the leveling screws, 
and the other half by the screw a. 

Repeat the operation until the bubble will settle in the middle of 
the tube after reversing the telescope. 

Third Adjustment. — The second adjustment being completed, re- 
volve the telescope in the y's, and if the bubble continues in the 
middle, the axis of the telescope and the axis of the tube CD lie in 
the same direction, or in the same vertical plane ; and if they be not 
in the same vertical plane the bubble will run to one end or the 
other ; in that case the side screw b will remedy the defect. 

The three adjustments are now made approximately, no one of 
them can be made perfectly while the instrument is greatly out of 
adjustment in relation to the others ; therefore commence anew. — 
Bring the bar EE over two of the leveling screws, and level the 
instrument; then turn it over the two other screws and level it in that 
direction also. Now, if we can turn the instrument quite round 



LEVELING. 



161 



without removing the bubble from the center it is in pretty good ad- 
justment, but if otherwise, as is to be expected, make all these 
adjustments over again; they can now be made with much Jess 
difficulty. 

It is important that a level should be in as perfect adjustment as 
possible, but perfection in all respects is almost, yea, quite impost • 
ble. Yet, with a level considerably out of adjustment, we can ob- 
tain the relative elevation of any two points, provided we can set the 
level midway between them. 

To illustrate this, suppose the instrument placed at I), midway 
between two perpendicular rods Aa Bb. 




Let ah represent the true horizontal line, but suppose that the in- 
strument is so imperfect, or out of adjustment, that when the level- 
ing tube CD is horizontal, the telescope would point out the rising 
line DA, and the rise would be Aa. On turning the instrument 
round and sighting to B, the rise must be the same as in the oppo- 
site direction : for the distance is the same, therefore A and B are as 
truly on a level with each other as a and b. 

By this problem, practical men complete the second adjustment of 
the instrument. They make the three adjustments as just explained, 
as accurately as possible. They then measure, very carefully 
the distance between two stations, as E and F t and set the in- 
strument exactly midway between them as represented in the last 
figure. 

They then level the instrument ( that is the tube CD ), and 
find the difference of the levels between E and F ( two pegs driven 
into the ground ). 

Now, suppose AE measures on the rod, - 4.752 feet. 

And BF " " 6.327 feet. 



Then E is above F - 
11 



1.575 feet. 



162 SURVEYING. 

They now bring the level near to one of the stations as E, and 
level it very accurately, and sight to the rod AE. 

Now, suppose the target stands at - - - 5.137 feet. 

To this add 1.575 feet. 

6.712~ 

The rod man now goes to the station F y puts his target on the rod 
exactly at 6.712, and the telescope is turned upon it, and the hori- 
zontal spider's line ought to just coincide with the target, and will, if 
the instrument is in perfect adjustment ; if it is not, the error is taken 
out by the screws n n. If the error was but slight, as in such cases 
it always is with good instruments, the adjustment is as complete as 
it can be made. 

With the level there must be 

A ROD. 

The rod is commonly ten feet long, and divided into tenths and 
hundredths, some have also a vernier scale which in effect subdivides 
to thousandths. The target slides up and down the rod, and car- 
ries the vernier on the back of the rod ; the target has equal alternate 
portions painted black and white for contrast. 

A party to take the necessary levels on the line of a railroad or 
canal, after the stations are measured off, should consist of a leveler, 
and assistant leveler, a rod man, and an axe man. 

The leveler and assistant leveler both keep book, and sometimes 
the rod man also. If there is no assistant leveler, the rod man will 
have an abundance of time to keep book, and under such circum- 
cumstances always does so. Under all circumstances, two persons 
keep book, to have a check on each other and guard against mistakes. 

In the field the aim is to put the level midway between the two 
stations, but they are not particular about it if the instrument is in 
good adjustment ; they rather take the most advantageous spot to 
sight from. 

When the ground admits of it, that is, sufficiently level, two or 
three intermediate stations are observed, as well as the extreme back 
and fore stations. The extreme back and fore stations are called 
changing stations ; at these stations pegs are driven into the ground 
by the axe man for the rod man to plant his rod, so as to secure 



LEVELING. 



163 



the same point for both the fore and back sight. At the intermedi- 
ate stations they have no pegs, and are not particular in any respect, 
for all errors cancel each other. 

The common railroad chain is 100 feet, divided into 100 links ; 
each link is therefore one foot. Levels are commonly taken at inter- 
vals of 200 feet, oftener if the ground is very uneven, but a station 
is considered 200 feet, and the number of the station multiplied by 
200 gives the number of feet from the commencement. 

The field book is kept thus : 

B. S. means back sight, F. S. fore sight, A. ascent, D. descent, T. total 
elevation above a common base. 

N. B. — When the back sight is less than the fore sight, the 
ground is descending, and the converse. 



Sta. 




B. S. 


F. S. 

7.21 


A. 


D. 


Total. 
100 

97.11 


4.32 




2.89 


1 


5.52 


8.17 




2.65 


94.46 


2 


*9.18 


6.27 


2.91 




97.37 


3 


6.27 


6.12 


0.15 




97.52 


4 


6.12 


3.76* 


2.36 




99.88 


5 


9.81 


11.62 




1.81 


98.07 


6 


8.47 


9.02 




0.55 


97.52 


7 


2.64 


8.91 




6.27 


91.25 


8 


1.07 


7.38 




6.31 


84.94 


9 


4.29 


5.32 




1.03 


83.91 


10 


5.32 


4.85 


0.47 




84.38 


11 


4.85 


3.17 


1.68 




86.06 


12 


8.22 


1.53 


7.31 




93.37 



Thus we go through the whole line. We commenced with the 
constant 100, but this is arbitrary; the object of taking a constant is 
to avoid the minus sign, that is, getting below our ruled paper in 
making a profile of the vertical section. 

Where the line is to be generally ascending, we assume a small 
constant, where generally descending a large one ; taking care in all 
cases to have it so large as not to run it out. 

At each and every section we know exactly how much we are 
above or below the constant base, and the exact ascent or descent 
from any one station to any other. 

The following diagram represents a vertical section of the ground 




11 10 



164 SURVEYING. 

where these levels were taken, with the exception of the exaggera- 
tion of the roughness caused by the difference of scales for the base 
and perpendicular lines. From one station to another is 200 feet ; 
we have made 10 feet occupy more space in the perpendicular direc- 
tion than 400 feet does in a horizontal direction. We do this to 
show more clearly where any particular grade will enter the 
ground, and how much it is necessary to cut or fill at any particular 
point. 

The zigzag line from 100 of 
altitude to 86, represents the 
surface of ground, and suppose 
that we wished to reduce it to 
a regular grade so as to remove 
as little earth as possible. By 
the mere exercise of judgment, 
we conclude to run the grade between station and 11, from 98 to 
92 ( but the propriety of this conclusion would depend on the con- 
tour of the ground on each side of these stations ). The direct line 
a b drawn, shows that the grade runs out of the ground at station 

I , we must fill in about 2^ feet at station 2, the grade runs into the 
ground again at about 80 feet before we come to station 3. At sec- 
tion 4 the cutting must be a little over 2 feet, at station 5 a little 
over 5 feet, at 7, 2 feet, and runs out of the ground midway be- 
tween stations 7 and 8. 

At stations 9 and 10 we must fill in about 8 feet, and so on ; the 
depth of cutting or filling is obvious at every station. 

If the contour of the ground beyond 1 1 was generally level or 
descending, we would change the grade at station 7 and render it 
more descending, so as to make less filling up at stations 9, 10, and 

II. In the adoption of grades for a railroad, an engineer has great 
scope to exercise his judgment. 

In the representation here made, ab appears like a steep grade, 
but it is not ; it could scarcely appear on the ground other than a 
level, for the difference is only 6 feet in 2200 feet of distance, which 
is at the rate of 14 r 4 ^ feet to the mile. 

Engineers can freely vary their grade, while it keeps under 1 8 or 
20 feet to the mile ; but they submit to a great deal of cutting and 
filling, before they establish a grade over 40 feet to the mile. 



LEVELING. 



165 



CONTOUR OF GROUND. 

Contour of ground is shown on maps, by marking where parallel 
planes run out on the surface. 

We shall give only the general principle. 

Let A be the top of a hill, whose contour we wish to delineate; 
measure any convenient line as AB, up or down the hill, and by 
the level or theodolite, ascertain the relative elevations of a, b, c, d, 
<fcc, as many planes as we wish to represent. 

At a, place the level or theodolite, and level it ready for observa- 
tion ; measure the height of the instrument, and put the target on 
the rod at that height. 

Send the rod-man and 
axe-man round the hill, on 
the same level as the in- 
strument. Let the rod-man 
set the rod ; the leveler 
will sight to it through the 
telescope, and if the tar- 
get is below the level, he 
will motion the rod-man up the hill, if too high, down the hill ; at 
length he will get the same level, and there the axe-man will drive 
a stake. In the same manner we will establish another stake further 
on ; and thus proceed from point to point. To get round the hill, 
it may be necessary to move the instrument several times. The 
plane thus established, is represented by the curve am. 

In the same manner, by placing the instrument at b, we can 
establish the next plane bn. 

Then the next, and the next, as many as we please. Where the 
hill is more steep, two of these parallel planes will be nearer together 
in the figure; where less steep, they will appear at a greater distance 
asunder, and this, with the proper shading, will give a true repre- 
sentation of the ground. 




166 SURVEYING. 

ELEVATIONS DETERMINED BY ATMOSPHERIC PRES- 
SURE, AS INDICATED BY T H E B AR O M E T E R . 

The higher we ascend above the level of the sea, the less is the 
atmospheric pressure (other circumstances being the same), and 
therefore we can determine the ascent, provided we can accurately 
measure the pressure, and know the law of its decrease. 

As this work is designed to be educational as well as practical, 
we shall here make an effort to explain the philosophy of the pro- 
blem, in such a manner, as to force it upon the comprehension of the 
learner. 

The pressure of the atmosphere at any place, is measured by the 
height of a column of mercury it sustains in the barometer tube. 

It is found by experiment, that air is compressible, 
and the amount of compression is always in pro- 
portion to the amount of the compressing force. 

]Now, suppose the atmosphere to be divided into 
an indefinite number of strata, of the same thick' 
ness, and so small that the density of each stratum 
may be considered as uniform. 

Commence at an indefinite distance above the 
surface of the earth, as at A, and let to represent 
the weight of the whole column of atmosphere 
resting on A. Let the small and indefinite dis- 
tances between AB, B C, CD, &c, be equal to each other, and we 
shall call them units of some unknown magnitude. 

The weight of the column of atmosphere supposed to rest on B, 
is greater than w, by some indefinite part of w, say the nth. part. 

Then the weight on B, must be expressed by( w-\-- )or ( n * \w. 

In the same manner, the weight or pressure resting on C, must be 
the weight above B, increased by its ?*th part ; that is, it must be 

( -—^— )w-\- (—-— } w > whichby addition is (^i_Vw. 

In the same manner, we find that the weight resting on D, must 
beL-L-U-w, and so on. For the sake of perspicuity, we recapitulate. 







A 
B 

D 



LEVELING. 167 

The pressure on A is w. Units from A 
" on B is ^^-tl\w " " 1 

on is fctii! 10 " "2 



(»+l) : 



on Z> is l"" 1 "^ 3 w " "3 

n 3 



(H-i)' 



" " on .# is 

" " on F is C^+i) 5 w " '< 5 

n 5 
&c. &c. &e. &c. 

Here we observe the series which represents the pressure of 
atmosphere, at the different points A,B, (7,&c, is a series in geometrical 
progression, and it corresponds with another series in arithmetical 
progression ; therefore, by the nature of logarithms, the numbers in 
the arithmetical series, may be taken as the logarithms of the num- 
bers in the geometrical series. 

But this system of logarithms, may not be hyperbolic nor tabular, 
indeed it is neither ; the base of this system is as yet unknown, but 
our investigations will soon lead to its discovery. 

Now, let the number of units from A to S (the surface of the 
sea), or to the lower of two stations, be represented by x, then the 

/ ' n~\~ 1 \ x 
expression for the pressure of the air would ber \ w, but this 

is neither more nor less than the weight of the column of mercury 
in the barometer, which is sustained by this pressure. 

By calling this b, and designating the logarithms of this unknown 
system by L ', we shall have 

L'b^x (1) 

Taking y to represent the number of units from A to V, and b t to 
represent the pressure of the air at that point, we shall have 

L'b,=y (2) 

Subtracting (2) from (1), we shall have 

L'b—L'b,=x—y=SV 
This is, a certain peculiar logarithm of the barometer column at the 
lower station, diminished by the logarithm of the barometer at the 



168 SURVEYING. 

upper station will give the difference of levels between the two sta- 
tions. But still all is indefinite and unknown, because we know 
nothing of these logarithms. 

In algebra, we learn that the logarithms of one system can be 
converted into another by multiplying them by a constant multiplier 
called the modulus of the system, therefore, 

Assume Z to be the modulus or constant that will convert com- 
mon logarithms into these peculiar logarithms. 

Then, Z(log. 6— log. b,)=SV (3) 

Here, log. b denotes the common logarithms of the barometer 
column. 

Equation (3) is general, and determines nothing until we know 
£Fin some particular case. 

Taking S V some known elevation, and observing the altitude of 
the barometer column at both stations, and then equation (3) will 
give Z once for all. 

Putting h to represent the known elevation, and we have, in general, 

h 
Z -\o g .b—log.b, ( 4 ) 

Example. — At the bottom and top of a tower, whose height was 

200 feet, the mercury stood in the barometer as follows. 

At the bottom, - 29.96 inches =b 

At the top, ----- 29.74 inches =b, 

the temperature of the air being 49° of Fahrenheit's thermometer. 

200 200 cocn „ , 

Whpnop Z= q ■ == =62500 nearly. 

vvnence, z, ^ 2996 __ log 29# 74— q.003201 j 

" But this multiplier is constant only when the mean temperature 
of the air at the two stations is the same ; and for a lower tempera- 
ture the multiplier is less, and for a higher it is greater. A cor- 
rection, however, may be applied for any deviation from an assumed 
temperature, by increasing or diminishing ( according as the tempera- 
ture is higher or lower) the approximate height by its 449th part for each 
degree of Fahrenheit's thermometer. We can moreover change the 
multiplier to a more convenient one by assuming such a tempera- 
ture as shall reduce this number to 60000 instead of 62500. Now 
62500 exceeds 60000 by its 25th part; and, since 1° causes a 
change of one 449th part, the proportion 

_!_ • 1 ° • ■ J_ • 1 7 Q 

449 ' l * ' 2 S * 1 '' V > 



LEVELING. 169 

gives 18° nearly for the reduction to be made in the temperature of 
the air at the time of the above observations, in order to change the 
constant multiplier from 62500 to 60000, or to 10000, by calling 
the height fathoms instead of feet. Thus, instead of the thermometer 
standing at 49°, we may suppose it to stand at 49° — 18° or 31°; and 
then, we take 10000 as the multiplier, and apply a correction addi- 
tive for the 18° excess of temperature." 

The same observations, for example, being given, to find the 
height of the tower. 

29.96 - - log. - - 1.47654 

29.74 - - - log. * - - 1.47334 

Diff. of log. - - 0.00320 

Multiplier - - - 10000 

Product 32 

Then the height of the tower is 32 fathoms, or 32X6 = 192 feet, 
on the supposition that the temperature of the air was 31° in place 
of 49°. But it being 49°, we must increase 192 by its T \j part for 
each degree above 31°, that is, by ^\ or jfe nearly of its approxi- 
mate height, which gives nearly 8 feet to add to 192, making 200 
feet for the height of the tower. 

The same method is applicable to other cases whatever be the 
temperature of the air at the two stations, provided it be the same 
or nearly the same at both stations, or provided we take the mean 
temperature of the two stations. We can find the difference of 
levels of two stations to considerable accuracy by the following 

Rule. — 1st. Take the difference of the logarithms of the two baro- 
meter columns, and remove the decimal 'point four places to the right. 
This is the approximate difference of levels in fathoms. 

2d. Add T | ¥ of the approximate height for each degree of tempera- 
ture above 31°, and subtract the same for each degree below 31° ; the 
result cannot be far from the truth. 

E X A MPLES. 

1. The barometer at the base of a mountain stood at 29.47 inches, 
and taking it to the top, it fell to 28.93 inches. 

The mean temperature of the air was 51°. What was the height 
of the mountain in feet ? Ans. 503.34 feet. 



170 SURVEYING. 

29.47 log. - - - 1.469380 
28.93 log. - - - 1.461348 

O008032 

Approximate height in fathoms, 80.32. 

Correction.— Add ffo of 80.32 to itself, that is, add 3.57. 
Height, in fathoms, - - 83.89 
Multiply by - - - - Q 

Height in feet, - - 503.34 

The average height of the barometer, at the level of the sea, is 
30.09 inches ; and now if we know the average height of the baro- 
meter at any other place, and the average temperature, it is equiva- 
lent to knowing the elevation of the latter place above the level of 
the sea. 

For example, the mean height of the barometer at Albany Academy 
is 29.96, and the mean temperature is 49°. How high is the 
academy above tide water ? 

30.09 log. 
29.96 log. 





Ans. 117.3 feet. 


1.478422 


49° 


1.476542 


31 


0.001880 


"l8° 


18.80 fathoms. 


75 





Approximate height 

Add j-Vk or fa ^ 

19.55X6 = 117.3 feet. 

2. The average height of the barometer at Amenia Seminary 
in Duchess, Co., New York, is stated in the Regents' report at 29.81 
inches : average temperature 49°. What is the height of that point 

above tide water ? 

Ans. 253.32 feet. 

3. The mean height of the barometer at Pompey Academy, On- 
ondagua Co., New York, is 28.13, corrected and reduced to 32° 
Fahr. What then is the elevation of that locality ? 

Ans. 1755 feet. 
Others have made it 1 745 feet. 

4. From various observations on the summit of Mount Washing- 
ton, in New Hampshire, the mean height of the barometer there is 
24.20, mean temperature at the times of observation was about 65° 
Fahr. 



LEVELING. 171 

Now admitting that the mean temperature at the surface of the 
sea, in the. same latitude must have been 75°, which would make 
the mean temperature between the two stations 70°, what then is 
the height of Mount Washington above the sea ? 

Ans. 6170 feet, nearly. 

By some observations the elevation is estimated at 6496 feet, by 
others at 6290 feet. 

5. Lieutenant* Fremont, in his narrative of the exploring expedi- 
tion across the Rocky Mountains, page 45, under date of July 1 3th, 

1842, states his latitude at 41° 8' 31", longitude 104° 39' 37", 
height of the barometer 24.86 inches, attached thermometer 68° ; 
what was his elevation above the sea ? Ans. 5389.2 feet. 

Remark. — The author states his elevation at 5449. feet. As he does not state 
the temperature of the air by the detached thermometer, we know not what 
correction he made. These solitary barometrical observations are more or less 
valuable, according to the settled or unsettled state of the weather. A person 
of experience and good judgment in such matters, like Fremont, would not of 
course record the inapplicable observations. 

6. Lieut. Fremont, in his journal, page 104, under date of August 
15th, 1842, when, as he supposed, on the highest point of the Rocky 
Mountains, observed the barometer to stand at 18.29 inches, and the 
attached thermometer at 44°-f ; what was the elevation above the 
level of the sea? Ans. 13522 feet. 

Fremont estimates the elevation at 13570 feet. 

This shows that he estimated the mean temperature above 50°, 
and no doubt a similar cause made the difference in the result of 
the previous example. 

7. On page 140, of Fremont's Journal, under date of July 12th, 

1843, he says ; " The evening was tolerably clear, with a tempera- 
ture at sunset, of 63°. Elevation of the camp, 7300 feet." 

Taking the mean temperature of the two stations, the sea and his 

* Lieutenant was his proper title at this time. 

f If the sea were at the base of the mountain, the temperature at the lower 
station would no doubt be as high as 60°. Making this supposition, the mean 
temperature of the two stations would be 50°. We therefore take 50° as the 
mean temperature. 



172 SURVEYING. 

place of observation, at 67°, what must have been the height of his 
barometer ? Ans. 23.21 inches. 

Represent the approximate elevation by y, then 

y+?^=7300 Or, y=6738.14 
* 449 ' : 

Divide y by 6, which gives 1126.35. Divide this by 10000. 
Then, let x represent the altitude of the barometer column. 

Whence, 1.478422— log. x=0. 11 2635 
Therefore, log. x= 1 .365787 

In the preceding examples we could only be general and approxi- 
mate, we had only the observations at one station referred to general 
observations at the other ; but our results cannot be far from the 
truth. 

"When the difference of temperatures at the two stations is con- 
siderable, the result must be affected by it. 

When the upper station is the coldest, which is generally the case, 
the mercurial column will be shorter than it otherwise would be, and 
consequently indicate too great a height. 

If the temperature of the upper station is taken for the tempera- 
ture of the lower, the mercurial column at the lower station would 
not be high enough, and the deduced result would be too small, as 
is the case in example 5. 

The contraction of mercury being about one 10000th part for 
each degree of cold, or, 0.0025 in a column of 25 in., it would 
require 4° difference of temperature, to produce an effect amounting 
to one division on the scale of a common barometer, where the 
graduation is to hundredths of an inch. 

This correction is combined with the former in the following 
equation, in which 1 1' represents the temperature of the air at the 
two stations ; t at the lower station, q and q' the temperature of the 
mercury, as indicated by the attached thermometer. 

The fraction 0.00223, is equal to T £ ¥ nearly ; A=the height 
sought, b and b t represent the observed height of the mercurial 
column. 

/*= 10000 jl +0.00223 f'-±-''— 3 At log. , , h —- - 

} \ 2 /) 8 1,(1+0.001(2— q') 



LEVELING. 173 

Beside the corrections previously considered, regard is sometimes 
had to the effect of the variation of gravity in different latitudes, 
and at different elevations above the earth's surface. The latter, 
however, is too small to require any notice in an elementary work. 
The former may be found by multiplying the approximate height 
by 0.0028371 Xcos. 2 lat. It is additive, when the latitude is less 
than 45°, and subtractive when greater. Or it may be taken from 
the following table. 



Latitude. 




Correction. 


0° - 


- 


' + ail of tlie a PP- height. 


5° 


- 


4- _!_ 

* 35 8 


10° - 


- 


- 4- _!_ 
' 3 7 5 


15° 


- 


4- _i_ 

^ 4 7 


20° - 


- 


"T" 460 


25° 


- 


T 548 


30° - 


- 


- 4- _i_ 

I 7 05 


35° 


- 


"T* To"3o 


40° - 


- 


. 4- — l — 

» 2 3 


45° 


- 





50° - 


- 


_ 1 

2 3 


55° 


- 


I 

10 3 


60° - 


- 


- — 1 

7 5 


65° 


- 


— 1 

54 8 


70° - 


- 


___ 1 

4 6 


75° 


- 


_ 1 

4 7 


80° - 


- 


- ___, 1 

3 7 5 


85° 


- 


— 1 

35 8 


90° - 


- 


_ I 

35 2 


Given, the pressure of th€ 


i atmosphere at the bottom of a moun- 


tain, equal to 29.68 in. of mercurj 


r , and that at its summit, equal to 


25.28 in., the mean temperat 


ure being 50°, to find the elevation. 




Ans. 


727.2 fathoms, or 4363.2 feet. 


The following observations being taken at the foot and summit of 


a mountain, namely, 






at the foot, bar. 29.862 attach, therm. 78° detach, therm. 71° 


at the summit, " 26.137 




« 63° " 65° 


to find the elevation. 








Ans. 


612.9 fathoms, or 3677.4 feet. 



174 SURVEYING. 

It is required to find the height of a mountain in latitude 30°, the 
observations with the barometer and thermometer being as follows ; 
namely, 

at the foot, bar. 29.40 attach, therm. 50° detach, therm.* 43° 
at the summit, " 25.19 " 46° " 39° 

Ans. 683.27 fathoms, or 4099.62 feet. 
If we assume any temperature, for instance 45°, and the height 
of the barometer at the level of the sea, at 30.09 inches ; we can 
compute the elevation of the point, where it would be 29.99, 29.89, 
29.79, 29.69, &c, inches ; and thus we might form a table, showing 
the elevations that would correspond to any assumed height of the 
barometer at that temperature. It will be found, that the first fall 
of T \ of an inch will correspond to about 88 feet in elevation, but 
every subsequent tenth, will require a greater and greater elevation. 

* The attached thermometer measures the temperature of the mercury in the 
barometer, and the detached thermometer, that of the surrounding air. 



NAVIGATION. 



CHAPTER I. 

INTRODUCTION. 

Navigation is the art of conducting a ship from one place to 
another. 

In most works this art is mixed up with seamanship and elementary- 
science. In this work, navigation will stand by itself — alone ; and 
we shall presume that the reader is properly prepared in elementary 
science. 

This being the case, it will not be necessary to take up time and 
space in giving definitions of latitude, longitude, meridian, horizon, 
&c, &c, the previous indispensible knowledge of geography neces- 
sarily gives a knowledge of all these terms. 

Navigation, rightly understood, requires an accurate knowledge 
of the geography of the seas — the winds and currents that here 
and there prevail, and also a good general knowledge of astronomy. 

Running a line in surveying and running a course at sea, are math- 
ematically the same thing, except that the latter is on a larger scale 
than the former, without its accuracy, and it is for a different object. 

In surveying we take no account of the magnitude and figure of 
the earth. In navigation we are compelled to do so, unless the 
limits of the operation be very small. 

There are two methods of finding the position of a ship. 

1st. By tracing her courses and distances, as in a survey. This 
is called dead reckoning. 

2d. By deducing latitude and longitude from observations on the 
heavenly bodies. This is called nautical astronomy. 

(175) 



176 NAVIGATION. 

No one expects accuracy from dead reckoning, and as a general 
thing it is only used from day to day, between observations ; or to 
keep the approximate run during cloudy weather or until observa- 
tions can again give a new starting point. 

Some navigators keep a continuous dead reckoning from port to 
port, which enables them to judge of drifts, currents, and unknown 
causes of error. 

The earth is so near a sphere that for the practical purposes of 
navigation it is taken as precisely so. Its magnitude corresponds 
to 69 l English miles to one degree of the circumference, but in the 
early days of navigation 60 miles were supposed to be about 
a decree, which for the sake of convenience is still retained. 

CD ' 

The sixtieth part of a degree is called a nautical mile, and it is, of 
course, larger than an English mile. 

In an English mile there are - 5280 feet. 

In a nautical mile there are - 6079 feet. 

The rate which a ship sails is measured by a line running off of 
a reel, called the log line. 

The log is nothing more than a piece of thin board in the form of 
a sector, of about six inches radius, the circular part is loaded with 
lead to make it stand perpendicular in the water. 

The line is so attached to it that the flat side of the log is kept 
toward the ship, that the resistance of the water against the face of 
the log may prevent it, as much as possible, from being dragged 
after the ship by the weight of the line or the friction of the reel. 

The time which is usually occupied in determining a ship's rate 
is half a minute, and the experiment for the purpose is generally 
made at the end of every hour, but in common merchantmen at the 
end of every second hour. As the time of operating is half a min- 
ute, or the hundred and twentieth part of an hour, if the line were 
divided into 120ths of a nautical mile, whatever number of those 
parts a ship might run in a half minute she would, at the same rate 
of sailing, run exactly a like number of miles in an hour. The 1 20th 
part of a mile is by seamen called a knot, and the knot is generally 
subdivided into smaller parts, called fathoms. Sometimes (and it is 
the most convenient method of division ) the knot is divided into ten 
parts, more frequently perhaps into eight ; but in either case the 
subdivision is called a fathom. 



INTRODUCTION. 177 

We shall consider a fathom the tenth of a knot, and as the nauti- 
cal mile is 6079 feet, the 120th part of it is 50.66, the length of a 
knot on the line, and a little over 5 feet is the length of a fathom. 

The operation of ascertaining the rate of sailing is called by sea- 
men heaving the log. 

At the end of an hour the loaded chip, or log, is thrown over the 
stern into the sea ; a quantity of the line, called the stray line, is 
allowed to run off, then the glass is turned and the number of knots 
that runs off the reel during the half minute is the rate of the ship's 
motion. 

The log is then hauled in and the same operation is repeated at 
the end of the next hour. 

The officer of the watch, who has been on deck during the hour, 
will mark on the slate or board, called the log board, the number 
of miles and parts of a mile which the ship has sailed during 
the last hour, according to the best of his judgment ; the log was 
thrown only to help make up that judgment, for the rate at the time 
the log was thrown may have been considerably more or less than 
the average motion during the hour. 

The course of a ship is marked by the mariner's compass. 

The mariner's compass differs from the surveyor's compass only 
by its construction, that is, the magnetic needle is the motive power 
in both. In consequence of the motion of a ship at sea, the mari- 
ner's compass is suspended in a double box, moving on a double 
axis, one at right angles with the other, the whole balanced by a 
central weight which keeps the compass card nearly steady and hori- 
zontal, whatever be the motion of the vessel. 

The card is attached to the needle, and is moved by the needle. 
The card is divided into 32 equal parts, called points, and to read 
over these points in order, is called by seamen, boxing the compass, 
and to know the north star and box the compass is too often the 
amount of the common sailor's knowledge of navigation. 
12 



178 



NAVIGATION. 




The figure 
before us re- 
presents the 
card of the 
mariner's 
compass. The 
four quadrant 
points are 
marked by a 
sino-le letter as 
2\ T . for north, 
E. for east. 
The midway 
points between 
these by two 
letters, as JV. 
E. for north- 
east, N. W. for north-west, &c. One point either way from any one 
of these eight points is marked by the word by. Thus, JV. by E. is 
one point from the north toward the east, and it is read north by 
east ; S. E. by & indicates one point from the south-east more to 
the south, and it is read south-east by south ; W. by N. means west 
one point toward the north. 

To box the compass we begin at any point, as north, and mention 
every point in order all the way round, thus : 

North; north by east; north north-east; north-east by north; north- 
east ; north-east by east ; east north-east ; east by north ; east, <fcc. 

A point of the compass is 11° 15', which is subdivided into four 
equal parts. Mariners never take into account a less angle than a 
quarter point, in running a course. 

When the mariner sets his course, be makes allowance for the 
variation of the needle, and his magnetic courses he reduces to true 
courses, by the following 

Rule. — Make a representation of the compass card on paper, and 
draw a line through the compass course. 

JTbw, conceive the compass card turned equal to the variation to the 
eastward, if variation is west, and vice versa. 



INTRODUCTION. 179 

The line will now pass over the true course. 

In the following examples, the tine courses are required. 



Compass Course. 


Variation. 


Answer. 
True Course. 


1. S. S.E.\E. 


2 J; W. 


S. E. b E. 


2. E. \ N. 


SE. 


E. S. E. \ S. 


3. N. W. b W. 


3|1; 


K b W. i W. 


4. W. S. W. i S. 


4 W. 


S. b W. | IT. 


5. S. S. W. 


liW- 


54 W. 


6. M 


5E. 


JS T .E.hE. 


7. E. b S. 


2iE. 


S. E. | E. 


8. S. 60°E. 


1S°W. 


S. 7Q°E. 


9. JV. 24 W. 


36 E. 


iV r . 12^. 


10. S. 16 W. 


40 E. 

LEEWAY. 


S. 56 W. 



The angle included between the direction of the fore and aft line 
of a ship, and that in which she moves through the water, is called 
the leeway. 

When the wind is on the right hand side of a ship, she is said to 
be on the starboard tack ; and when on the left hand side, she is said 
to be on the larboard tack ; and when she sails as near the wind as 
she will lie, she is said to be dose haided. Few large vessels will lie 
within less than six points to the wind, though small ones will some- 
times lie within about five points, or even less ; but, under such 
circumstances, the real course of a ship is seldom precisely in the 
direction of her head ; for a considerable portion of the force of the 
wind is then exerted in driving her to leeward, and hence her course 
through the water, is in general found to be leeward of that on which 
she is steered by the compass. Therefore, to determine the point 
toward which a ship is actually moving, the leeway must be allowed 
from the wind, or toward the right of her apparent course, when she 
is on the larboard tack ; but toward the left, when she is on the star- 
hoard tack. 

It is only when a ship crowds to the wind, that leeway is made. 

It is seldom that two ships on the same course make precisely the 
same leeway ; and it not unfrequently happens, that the same ship 
makes a different leeway on each tack. It is the duty of the officer 



180 NAVIGATION. 

of the watch, to exercise his best skill in determining, or estimating 
how much this deviation from the apparent course amounts to ; and 
in the dark, the chief reliance must be placed on the judgment of the 
experienced mariner. 



C HAP TER II. 

PLANE SAILING. 

In plane sailing, the earth is considered as a plane, the meridians 
as parallel straight lines, and the parallels of latitude as lines cut- 
ting the meridians at right angles. And though it is not strictly- 
correct to consider any part of the earth's surface as a plane, yet 
when the operations to be performed are confined within the distance 
of a few miles, no material error will arise, from considering them 
as performed on a plane surface. And, as we have already seen, in 
all questions where the nautical distance, difference of latitude, de- 
parture, and course, are the objects of consideration, the results will 
be the same, whether the lines are considered as curves drawn on 
the surface of the globe, or as equal straight lines drawn on a plane. 

In all maps, and charts, and constructions, when it is not other- 
wise stated, it is customary to consider the top of the page as pointing 
toward the north, the lower part as the south, the right side as the 
east, and the left as the west. The meridians therefore, in any 
construction, will be represented by vertical lines, and the parallels 
of latitude, by horizontal ones. 

Hence, in constructing a figure for the solution of any case in 
plane sailing, the difference of latitude will be represented by a ver- 
tical line, the departure by a horizontal one, and the distance by the 
hypotenusal line, which forms, with the difference of latitude and 
departure, a right-angled triangle ; and the course will be the angle 
included between the difference of latitude and distance. 

With this understanding, the solution of any case that can arise 
from varying the data in plane sailing will present no difficulty. 



PLANE SAILING 



181 




EXAMPLES. 

If a ship sail from Cape St. Vincent, Portugal (Lat. 37° 2' 54" 
north), S. W. j S. 148 miles ; required her latitude in, and the 
departure which she has made ? 

Bij Construction. — Draw the vertical line AB, to represent 
the meridian ; from the point A, make the 
angle BAC=S^ points, the given course ; and 
from a scale of equal parts, take A (7= 148 miles, 
the given distance ; from C on AB, draw the 
perpendicular CB, then AB will be the dif- 
ference of latitude, and BC the required de- 
parture, and measured on the scale from which 
AC was taken, AB will be found 114.4, and 
BC93.9. 

Lat. left --- 37° 2' 54" W. 

Diff. lat. - - - - 1 54 24 j\ T . 

Lat. in - - - - 35° 8' 30 7 JY. Dep. 93.9 

2. If a ship sail from Oporto Bar, in Lat. 41° 9' north, K W. %W. 
315 miles ; required her departure and the latitude arrived at ? 

Ans. Dep. 233.4 miles W.; Lat. 44° 41' JUt. 

3. If a ship sail from lat. 55° 1' K, S. E. by S., till her depart- 
ure is 45 miles ; required the distance she has sailed and her 
latitude ? Ans. Dis. 81 miles ; Lat. 53° 54' K 

4. A ship from lat. 36° 12' A r ., sails south-westward, until she 
arrives at lat. 35° 1' JV. t having made 76 miles departure ; required 
her course and distance. 

Ans. Course S. 46° 57' W. ; Distance 104 miles. 

5. If a ship sail from Halifax, in lat. 44° 44' K, S. E. \E., until 
her departure is 128 miles ; required her latitude and distance sailed. 

Ans. Lat. 42° 51' K, and dis. sailed 165.6 miles. 

6. A ship leaving Charleston light, in latitude 32° 43' 30" north, 
sails N. eastward 128 miles, and is then, by observation, found 39 
miles north of the light ; required her course, latitude, and departure. 
Ans. Lat. 33° 22' 30" N. ; Course IP. 72° 16' E. ; Dep. 122 miles. 

7. A ship from Cape St. Roque, Brazil, in latitude 5° 10' south, 
sails iV. E. \ iV!, 7 miles an hour, from 3 P. M. until 10 A. M. ; 
required her distance, departure, and latitude in. 

Ans. Dis. 133. miles ; Dep. 84.4 miles ; Lat. in, 3° 27' south. 



182 



NAVIGATION. 



8. A ship from latitude 41° 2' M, sails JY. K W. f W. 5|- miles 
an hour, for 2-|- days ; required her distance, departure, and latitude 
arrived at. 

Ans. Dis. 330 miles ; Dep. 169.7 miles ; Lat. 45° 45' K 

Similar examples, might be given without end, but these are 
sufficient, for they only involve the principles of the solution of a 
plane right-angled triangle. 

In the preceding examples it will be observed that we traced lat- 
itude from latitude, and the distances east and west we called de- 
parture — not difference of longitude. It now remains to determine 
difference of longitude from departure. 

On the equator, 60 miles of departure are equal to one degree of 
longitude, and the further we are from the equator, north or south, 
that is, the greater the latitude we are in, the same departure will 
cover more longitude. 

To discover the law for changing departure to difference of longi- 
tude, we adduce the following figure. 

Let C be the center of the earth, P the 
pole, PC a portion of the earth's axis. 

The plane PCB is the plane of one 
meridian, and P CA the plane of another 
meridian. 

AB is a portion of the equator between 
the two meridians. A CB is a sector in 
the plane of the equator, and DEH and 
EGQ are sectors similar to ACB. 

Observe that DE, EG, &c, are parallels of latitude, that is, they 
are parallel to AB, the plane of the equator. 

The magnitude of DE is called departure, and it corresponds to 




the difference of lonq\ 



AB. 



Also, EG is departure corresponding to the same difference of 
longitude AB. The difference of longitude is always greater than 
any corresponding departure, that is, AB is obviously greater than 
any oilier parallel distance between the same two meridians. 



PLANE SAILING. 183 

Because the two sectors A CB DEH are similar, we have the 
proportion. 

AC : DH : : AB : DE (1) 

Observe that A C is the radius of the sphere, DH is the sine of 
the arc PD, or the cosine of DA, which is the cosine of the lat- 
itude of the point D. 

Therefore the preceding proportion becomes, 

rad. : cos. lat. : : dif. Ion. : dep. 
Or, cos. lat. : rad. : : dep. : dif. Ion. 

In words, The cosine of the latitude is to the radius so is the depar- 
ture to the difference of longitude. 

These words are indelibly engraved on the memory of every nav- 
igator, and they embrace all the rules that can be made for chang- 
ing departure into longitude, or longitude into departure. 

When a ship sails east or west, the distance sailed is called de- 
parture, and is reduced to longitude by the preceding proportion. 
This is called parallel sailing. 

EXAMPLES. 

1. What difference of longitude corresponds to 47 miles departure 

in the latitude of 37° 23' ? Ans. 59.15 miles. 

Let x= the difference of longitude required. 

, 47X rad - 

Then cos. 37° 53' : rad. : : 47 : x= ^rr^r 

cos. 37° 23 

By log. 47 R - - - 11.6720.98 

Cos. 37° 23 - - - 9.900144 



Diff. Lon. 59.15 - - - 1.771954 

2. How many miles, or how much departure corresponds to a 

degree in longitude on the parallel of 42° of latitude ? 

Ans. 44.59 miles. 

Here the longitude of one degree is given. 

«■'.■■„ 60 cos. 42° 
B. : cos. 42° : : 60 : x= -^ 

By log. 60 ... 1.778151 
cos. 42° - 9.871073 



44.59 .... 1.649224 



Cape •! 



184 NAVIGATION. 

N. B. — In this manner the length of a degree in longitude cor- 
responding to each degree of latitude has been found and put in a 
table. 

3. A ship sails east from Cape Race, 212 miles; required her long- 
itude. The latitude of the cape is 46° 40' K, longitude 53° 3' 15" 
west. Ans. Ion. 47° 54' west. 

4. Two places in lat. 50° 12' differ in longitude 34° 48'; required 
their distance asunder in miles. Ans. 1336. 

5. How far must a ship sail W. from the Cape of Good Hope, that 
her course to Jamestown, St. Helena, may be due north ? 

Ans. 1193 miles. 

lat. 34° 29' S. T ' j lat. 15° 55' S. 

Ion. 18° 23' E. Jamestown j lon> 50 43 , 30 „ ^ 

6. How far must a ship sail E. from Cape Horn to reach the meri- 
dian of the Cape of Good Hope ? The latitude of Cape Horn 
being 55° 58' 30" &., Ion. 67° 21' W. y and the latitude and longi- 
tude of the Cape of Good Hope being as stated in the last example. 

Ans. 2878 miles. 

7. In what latitude will the difference of longitude be three times 
its corresponding departure ? In other words in what latitude will 
FG be one-third of AB ? ( See last figure.) 

Ans. lat. 70° 31' 44". 

8. Take the 2d example in plane sailing ( page 181 ), the depar- 
ture made, as stated in the answer, is 233.4 miles. What is the 
corresponding difference of longitude ? Ans. 5° 18' 24". 

This inquiry now arises. To what latitude does this departure 
correspond? Is it to the latitude left, 41° 9', or to the latitude ar- 
rived at, 44° 41' ? Or does it correspond to the mean latitude be- 
tween the two ? 

If we suppose the departure corresponds to lat. 41° 9', then the 
difference of longitude by the preceding rule must be 5° 10', and if 
the departure corresponds to lat. 44° 41', then the difference of lon- 
gitude is 5° 28'; the mean of these is 5° 19', and if we take the de- 
parture to correspond with the mean latitude 42° 55', then the dif- 
ference of longitude would be 5° 18' 24". 

In the examples under Plane Sailing we have supposed the earth 
a plane, and the course a ship sails a straight line, but neither sup- 
position is strictly true. 




PLANE SAILING. 185 

Meridians are not parallel with each other, and therefore when a 
ship sails by the compass, and cuts all the meridians at the same 
angle, the line that the ship sails will not be a right line : it will be 
a curve line peculiar to itself, called a rhumb line. 

For the sake of illustration, let us suppose 
that in the annexed figure, P is the north 
pole, KQ the equator, or a great circle, 
every part of which is a quadrant distance 
from P ; PK, PL, PM, &c, great circles 
passing through P, and of course cutting 
the equator at right angles ; AT, bB, PS, 
&c, arcs of smaller circles parallel to the 
equator, and therefore cutting the meridians 
at right angles ; AE a curve cutting every 
meridian which it meets, as PK, PL, PM, &c, at the same angle. 
Then PK, PL, &c, produced till they meet at the opposite pole, 
are called meridians ; AT, bB, PS, <fec, continued round the globe, 
are called parallels of latitude ; AE is called the rhumb line, passing 
through A and E ; the length of AE is called the nautical distance 
from A to E ; and the angle BAb, or any of its equals, cBC, dCD, 
&c, is called the course from A to E. 

If a ship sail from A to E, EF will be her meridian distance ; but 
if she sail from E to A, AI will be her meridian distance. 

If AB, BC, CD, &c, be conceived to be equal, and indefinitely 
small, and their number indefinitely great ; then the triangles A Bb, 
BcC, &c.,may be considered as indefinitely small right-angled plane 
triangles. And as the angles BAb, CBc, &c, are equal, and the 
right angles AbB, BcC, &c, are equal, the remaining angles ABb, 
BCc, &c, are equal ; and as the sides AB, BC, &c, are also equal, 
these elementary triangles ABb, BCc, CBd, &c, will be all iden- 
tical triangles ; therefore AE will be the same multiple of AB, that 
the sum of Ab, Be, Cd, &c, is of Ab ; and that the sum of Bb, Cc, 
Dd, &c, is of Bb. 

It is obvious that 

Ab-\-Bc+Cd-\-&c.=AF 
the whole difference of latitude. And, 

Bb-{- Cc-\-Pd-\-Ee-{-&c.=t'he whole departure. 



188 NAVIGATION. 

But this departure is neither EF nor AI, it is greater than EF 
and less than AI; because IB is less than its corresponding por- 
tion on AI and greater than its corresponding portion on FE. The 
same may be said of cC, Dd, &c. Therefore the departure on any 
course corresponds to neither of the extreme latitudes, but to some mean 
between the two, and it is so near the arithmetical mean, that the 
arithmetical mean is taken as the true. 

Therefore in all those examples in Plane Sailing, on page 181, we 
can take the departures and find the corresponding differences of 
longitude, provided we take the middle latitude and consider the 
departure run on that parallel. 

This method of connecting the change in longitude with a ship's 
change of place is called 

MIDDLE LATITUDE SAILING. 

But in reality there is no such thing as middle latitude sailing ; 
the cosine of the middle latitude is compared to the radius, as the 
ratio between the departure made and the corresponding difference 
of longitude, but the departure made may be made on one course or 
on several courses. When a ship sails on several courses before 
the run is summed up, the summing up and finding the result in one 
course and distance is called working a traverse, and sailing from one 
point to another by several courses is called 

TRAVERSE SAILING. 

With adverse winds or crooked channels, vessels are obliged to 
run a traverse. Going round a survey and keeping an account of 
our course and distance from the starting point is working a traverse, 
and the operation is the same on sea or land, except on land we aim 
at coming round to the same point again, but on sea we wish to make 
some other point. 

With this explanation it is obvious that we must make a table as 
in a survey, and compute the course and distance from the starting 
point, and this is called the course and distance made good. 

To work a traverse we use the traverse table of course ; that table 
is made to every half degree, and the column in the table nearest to 
the course is sufficient! v exact. 



TRAVERSE SAILING. 



187 



The following table gives the degree and parts of a degree corres- 
ponding to every point and quarter point of the compass. 



Points. 


Deg. 


Points. 


Dog. 


i 


2° 48' 45" 


H 


47° 48' 45" 


i 


5° 37' 30" 


H 


50° 37' 30" 


1 


8° 26' 15" 


4f 


53° 26' 15" 


1 


11° 15' 


5 


56° 15' 


i-l 


14° 3' 45" 


H 


59° 3' 45" 


H 


16° 52' 30" 


H 


61° 52' 30" 


if 


19° 41' 15" 


5f 


64° 41' 15" 


2 


22° 30' 


6 


67° 30' 


2* 


25° 18' 45" 


6* 


70° 18' 45" 


n 


28° 1' 30'' 


6* 


73° 7' 30" 


H 


30° 56' 15" 


6| 


75° 56' 15' 


3 


33° 45' 


7 


78° 45' 


3± 


36° 33' 45" 


*i 


81° 33' 45" 


» 


39° 22' 30" 


n 


84° 22' 30" 


- 


42° 11' 15" 


n 


87° 11' 15" 




45° 0' 


8 


90° 0' 



3f 



In works exclusively designed for practical navigation, the traverse 
table is adapted exactly to the points and quarter points of the 
compass, but the table in this work is sufficient for the purpose. 

The use of this table is to find the degree corresponding to any 
given course, thus : JV. by E., K by W., S. by E., S. by W., 
each correspond to 1 point or 11° 15'. In using our traverse table 
for 1 point we should take a mean result between 11° and 11° 30', 
which mean result can be taken by the eye. 

Again S.E. by E. \ E. is 5^ points, or S. 59° E. nearly, and so on 
for any other course that may be named. 

The student is now fully prepared to work the following examples 
in traverse sailing. 

EXAMPLES. 



1. A ship from Cape Clear, Ireland, in lat. 51° 25' JY. and longitude 
9° 29' W., sails as follows : 

S. S. E. I E. 16 miles, E. S. E. 23 miles, S. W. by W. -i W. 36 
miles, W. f ^\ r . 12 miles, and S. E. by E. I E. 41 miles. 



188 NAVIGATION. 

Required her course and distance made good, the departure, lati- 
tude and longitude of the ship. 

By Construction. — Take A for the place sailed from, and draw 
the vertical line NASC, to represent the meridian. About A, as a 
center, with the chord of 60°, 
describe a circle, cutting NC in 
iVand 8; then N and S will 
represent the north and south 
points of the compass. Take 2 J 
points from the line of chords,* 
and apply it from S to a, join 
Aa, and on it take AD= 16 from 
a line of equal parts. Then D 
will be the place of the ship at 
the end of the first course. From 
S, set off Sb—6 points from the 
line of chords ; join A b, and 
through D draw BE, parallel to 
Ab, and make it equal to 23 from 
the same scale of equal parts 
that AD was taken from. Then 
E will be the place of the ship, 
at the end of the second course. Make Sc=5^ points, Nd 7£ 
points, and Se 5± points, taken from the line of chords. Through 
E draw EF, parallel to Ac, and make it equal to 36, from the scale 
of equal parts ; through F draw FO parallel to Ad, and make it 
equal to 12, from the scale of equal parts ; through G draw GB 
parallel to Ae, and make it equal to 41, from the scale of equal 
parts. Demit BO a perpendicular, on the meridian NC, and join 
AB ; then B will be the place of the ship, AB her distance from 
the place which she left, A her difference of latitude, B C her de- 
parture, and BA C the course whicli she has made on the whole. 
Now, AB, AC, and BC being measured on the scale of equal parts 
from which the distances were taken, we have ^4i?=62.7, AC= 
59.6, and BC=\9.6 miles. And the arc included by A C and AB 
if measured on the line of chords, gives about 18° for the measure 
of the course BA C. 

* Two and | points is 25° 18' 45", that is, take the chord of 25° 18' in the 
dividers, and set it off from s to a, and so on, for other angles. 




TRAVERSE SAILING. 



189 



TRAVERSE TABLE, 



Courses. 


Points. 


Dis. 


Diff. Lat. Dep. 




H 


16 


N. 


s. 


E. 


w. 


& S.E.±E. 


14.5 


6.8 


E. S. E. 


6 


23 




8.8 


21.3 




S. W.by W.1W. 


5^ 


36 




17.9 




31.8 


W.%& 


H 


12 


1.8 






11.9 


S.E.byE.^E. 


H 


41 




21.1 


35.2 








1.8 


61.4 


63.3 


45.7 










1.8 


43.7 





Result 59.6 | 19.6 

Lat. left - - 51° 25' JT. 
diff. lat. - - - 1 00 S. 


Lat. in - - 50 
To find the course and distance, by tr 
As dif. lat. 59.6 miles 

: radius 90° 
: : dep. 19.6 miles 


25 JF". Mid.lat. 5( 
igonometry, 
1.775246 
10.000000 
1.292256 


: tan. course 18° 12' 

As sin. course 18° 12' 
: dep. 19.6 miles 
: Radius 


9.517010 

9.484621 

1.292256 

10.000000 


: Dis. 62.75 miles 
To find the dif. of longitude. 
As cos. 50° 55' 

: Radius 
: : dep. 19.6 


1.797635 

9.799651 

10.000000 

1.292256 


: diff. Ion. 31.09 miles 

Longitude left 
diff. Ion. 


1.492605 

9° 29' west 
31 east 


Lon. in 


8° 58' west 



Thus, we have found the course 18° 12' ;• Distance 62.75 miles ; 
diff. longitude 31' E.; lat. in 50.25 N.\ lon. 8° 58' W. 

If these be the distances run in a day, from noon to noon again, 
then the preceding operation is called working a day's work ; other- 
wise it is called working a traverse, as we have mentioned before. 



190 NAVIGATION. 

But this is not the seaman's way of working a day's work, he does 
it all by inspection, in the traverse table. For example, taking the 
result of the traverse 59.6 south, and 19.6 east, which shows that 
the resulting course is between the south and the east, and with 
these numbers he enters the traverse table, and finds, as near as 
possible, 59.6 and 19.6, standing as latitude and departure ; and 
they are found nearly under the angle of 18°, and opposite the dis- 
tance 63. nearly. 

Hence, he takes his course as S. 18° E., and dis. 63. To find 
the difference of longitude, he takes the middle latitude as a course, 
and the departure as difference of latitude, then the distance in the 
table is difference of longitude. 

In this instance, we take 51° as a course, and in the difference of 
latitude column we find 19.5, and the distance opposite to it is 31., 
which we take for difference of longitude. 

The reason for this is as follows : 

For the longitude we have, 

cos. mid. L : R : : dep. : diff. Ion. (1) 

In the construction of the traverse table, we have, 

cos. course : JR. : : diff. lat. : dist. , (2) 

Now, in proportion (2), if we take the middle latitude for a course, 
and the dep. for difference of latitude ; it necessarily follows, th at 
the last term of proportion (2) must be diff. of longitude; for 
proportion (2) would then be transformed into proportion (1). 

2. A ship sails from Cape Clear, as follows ; S. by W. 23 miles ; 
W. S. W. 40 miles ; S. W. f W. 1 8 miles ; W. \ N. 28 miles ; S. by 
E. 12 miles ; S. S. E.%E.\Q miles. 

Required the course and distance made good, and the latitude 
and longitude arrived at. 

Ans. Course S. 45° 47' W. ; dis. 102.4 miles. 
Latitude of ship 50° 14' F. ; Lai. 11° 25' W. 

3. A ship at noon, on a certain day, was in lat. 41° 12' N., and 
longitude 37° 21' W., she then sailed as follows : 

S. W. by W. 21 m. ; S. W. £ S. 31 m. ; W. S. W. % S. 16 m. ; 
S.^E. 18 m.; S. W. I W. 14, and W. | K 30 miles. 

Required her course, distance, latitude, and longitude. 
Ans. course S. 52° 49' W.\ dis. 111.7 ; lat. 40° 5' N. ; Ion. 39° 18' W. 



SAILING IN CURRENTS. 191 

4. Last noon we were in latitude 28° 46' south, and longitude 
32° 20' West ; since then we have sailed by the log : 

5. W.% W. 62 m. ; S. by W. 16 m. ; W. £ S. 40 m. ; S.W.^W. 
29 m. ; S. by E. 30 m. ; and S. f E. 14 miles. Required the direct 
course and distance, and our present latitude and longitude. 

Ans. Course S. 43° 14' W. ; Dis. 158 m. ; lat. 30° 41' S.; 

Ion. 34° 24' W. 
5. A ship from Toulon, lat. 43° 7' K, Ion. 5° 56, E., sailed 

5. S. W. 48 m. ; S. by E. 34 m. ; S. W. I W. 26 m. ; and E. 
17 miles. Required her course and distance to Port Mahon, Lat. 
39° 52' N., and longitude 4° 18 30" east. 

Ans. Lat. of ship 41° 32' K ; Ion. 5° 37' east. 
Course to Port M. S. 31° W. nearly, and distance 117.5 miles. 

6. On leaving the Cape of Good Hope, for St. Helena, we took 
our departure from Cape Town, bearing S. E.bj S. 12 miles, after 
running J\ T . W. 36 m., and N. W. by W. 140 miles. Required our 
latitude and longitude, and the course and distance made. 

N. B. Lat. of Cape Town 33° 56' S. Lon. 18° 23' E. 

Lat. of St. Helena 15° 55' S. Lon. 5° 43' 30" W. 

Ans. Lat. 32° 3' S. ; Lon. 15° 25' E. ; course ¥. 52° 41' W. ; 

dis. 187 miles. 

SAILING IN CURRENTS. 

If a ship at B, sailing in the direction BA, were in a current which 
would carry her from B to C, in the same time that in still water she 
would sail from B to J, then, by the joint 
action of the current and the wind, she would 
in the same time, describe the diagonal BD 
of the parallelogram ABCD. For her being 
carried by the current in a direction parallel 
to BC, would neither alter the force of the 
wind, nor the position of the ship,northe sails, with respect to it ; the 
wind would therefore continue to propel the ship in a direction 
parallel to AB, in the manner as if the current had no existence. 
Hence, as she would be swept to the line CD, by the independent 
action of the current, in the same time that she reached the line AB, 
by the independent action of the wind on her sails, she would be 
found at D, the point of intersection of the lines AD and CD, hav- 
ing moved along the diagonal BD. 




192 NAVIGATION. 

]STow the log heaved from the ship in the ordinary way, can give 
no imitation of a current ; for the line withdrawn from the reel is 
only the measure of what the ship sails from the log ; and, conse- 
quently, as the log itself, as well as the ship, will move with the 
current, the distance shown by the log in a current, is merely what 
it would have been if the ship had been in still water. 

If the ship sail in the direction of the current, the whole effect of 
the current will be to increase the distance ; but if she sail against 
the current, the difference between the rate of sailing given by the 
log and drift of the current, will be the distance which the ship 
actually goes ; and she will move forward, if her rate of sailing be 
greater than the drift of the current, but otherwise, her motion will 
be retrograde, or she will be carried backwards, in the direction of 
the current. 

Problems relating to the oblique action of a current upon a ship, 
may be resolved by the solution of an oblique-angled plane triangle, 
such as ABD, in the preceding figure, where if AB represent the 
distance which a ship would sail in still water, and AD the drift of 
the current in the same time, BD will be the actual distance sailed, 
and ABD the change in the course produced by the current. 

A great variety of problems might be proposed relative to currents, 
but the chief ones of any practical importance, are the following : 

1. To determine a ship's actual course and distance in a current, 
when her course and distance by the compass and the log, and the 
setting and drift of the current, are given. 

2. To find the course to be steered through a known current, the 
required course in still water, and the ship's rate of sailing, being 
known. 

3. To find the setting and drift of a current, from a ship's actual 
place, compared with that deduced from the compass and the log. 

The first of these cases may be conveniently resolved, by con- 
sidering the ship as having performed a traverse, the setting and 
drift of the current being taken as a separate course and distance. 

EXAMPLES. 

1. If a ship sail W. 28 miles in a current, which in the same time 
carries her ffl. K W. 8 miles, required her true course and distance. 



SAILING IN CURRENTS. 



193 



N. B. Conceive the current to be one course and distance, and 
with the other courses find the course and distance made good. 
Thus, by the traverse table : 



Course. 


Dis. 

28 


Diff. Lat 

N. S. 


Dep. 
E. W. 


w. 








28 


jsr. ir. w. 


8 


7.39 
7.39 






3.06 
31.06 



As 7.39 : rad. :: 31.06 : tan. 76° 36', the course, 
cos. 76° 31' : R : : 31.06 : 31.93, the distance. 

2. If a ship sail E. 7 miles an hour by the log, in a current setting 
E. N. E. 2.5 miles per hour ; required her true course, and hourly 
rate of sailing. 

Ans. Course K 84° 8' E., and rate 9.358 per hour.- 

3. A ship has made by the reckoning N. \ W. 20 miles, but by 
observation it is found, that, owing to a current, she has actually 
gone JfiF. iV". E. 28 miles. Required the setting and drift of the cur- 
rent in the time which the ship has been running. 

Ans. Setting & 64° 48' E., and drift 14.1 miles. 

4. A ship's course to her port is W. N. W., and she is running 
by the log 8 miles an hour, but meeting with a current setting 

W. -J- S. 4 miles an hour, what course must she steer in the current 
that her true course may be W. iV. PT? 

Ans. Course 2jF. 44° 39' W. 

5. In a tide running N. W.b W. 3 miles an hour, I wished to 
weather a point of land, which bore N". E. 14 miles. What course 
must I steer so as to clear the point, the ship sailing 7 miles an hour 
by the log, and what time shall I be in reaching the point ? 

Ans. Course N. 69° 51' E., and time 2 hours 25 minutes. 

6. From a ship in a current, steering W. S. W. 6 miles an hour 
by the log, a rock was seen at 6 in the evening, bearing S. W. -J S. 
20 miles. The ship was lost on the rock at 1 1 P. M. Required the 
setting and drift of the current. 

Ans. Setting S. 75° 10' J3. t and drift 3.11 miles per. hour. 
13 



194 



NAVIGATION. 

CHAPTER III 



MERCATOR'S CHART AND MERCATOR'S 
SAILING. 

In representing any small portion of the earth's surface, it is suf- 
ficiently accurate to represent the meridians as parallel ; but if the 
portion of the earth is considerable, the representation will not be 
true unless the meridians are curved. 

If we make a chart and draw all the meridians parallel with each 
other, the length of a degree of longitude in all places, except on the 
equator, will be greater on the chart than its true distance, but the 
true bearing of one place from another will be preserved, provided 
we increase the degrees of latitude in the same ratio as the degrees 
of longitude are increased. 

Gerrard Mercator, a Fleming, in 1556, published a chart which 
seemed to embrace this idea, but he did not show its construction, 
nor were his degrees in their true proportion ; but from this came 
the name of Mercator's Chart. 

A Mr. "Wright, an Englishman, in 1599, it is said, published the 
true sea chart, constructed on the following principles. 

1 . The distance between two meridians at the equator, is to their dis- 
tance in any parallel of latitude, as the radius is to the cosine of that 
latitude. 

2. Any part of a parallel of latitude, is to a like part of the meri- 
dian, as the radius is to the secant of that parallel. 

We shall make an effort to illustrate these principles by the fol- 
lowing figure. 

Conceive the equator to be extended both ways parallel to the 
earth's axis, thus forming a cylinder, whose circumference is just 
equal to the circumference of the earth. 




MERCATOR'S CHART 



195 



Let Qq be the plane of the equator, Pp the earth's axis ; con- 
ceive a globe enclosed in the cylinder, HLNM. 

Suppose there is an island on the earth at a, that island is projected 
on the cylinder at A. The surface of the earth at b is projected at 
B. Conceive this paper cylinder cut by a line at right angles to 
the equator and rolled out, it will then be a true representation of 
Mercator's chart. 

The scale on the globe at a is, to the scale on the chart at A, as 
Ca to CA, that is, as radius to the secant of the latitude at a. 

The scale on the chart at A is, to the scale on the chart at B, as 
CA is to CB, that is, the scale on the chart increases as the secants 
of the latitudes increase. 

The poles of the earth, and places very near the poles, can never 
be represented on this chart. 

The meridian distance of a degree on the globe, as at a, is 60 
miles, on the chart at A it is 60, into A C the secant of the latitude, 
calling Ca unity. 

If we commence at the equator at Q, and take one mile for 
unity. Then, 

Mer. pts. of 1'= nat. sec. 1 
Mer. pts. of 2'= nat. sec. 1-f- nat. sec. 2 
Mer. pts. of 3'= nat. sec. l'-f- nat. sec. 2'-}- nat. sec. 3 
Mer. pts. of 4'= nat. sec. l'-f- nat. sec. 2'+ nat. sec. 3' 
+ nat. sec. 4', &c, &c. 
In this manner the table of meridional parts was originally con- 
structed. It is Table IV of this work. 

The following figures represent any problem than can arise in 
Mercator's sailing. 

A C represents the true dif- 
ference of latitude. 

AD represents the meridion- 
al difference of latitude, which 
is always taken from the table. 

CB represents the departure. 

BE the difference of longi- 
tude. 




196 NAVIGATION. 

AB represents the distance. 
A, the angle at A y represents the course. 
Three of these six quantities must be given to solve a problem. 
Observe that the difference of longitude DE is always greater 
than the departure CB, as it ought to be. 

EXAMPLES. 

1. A ship from Cape Finisterre, in lat. 42° 56' JST., and longitude 
8° 16' W. y sailed S. W. \ W. till her difference of longitude is 134 
miles ; required the distance sailed and the latitude in. 

By logarithms. As radius ------ 10.000000 

: diff. Ion. 134 miles - - - - 2.127105 

: : cot. course 4j- points - - - 9.957295 
: mer. diff. lat. 121.5 miles - - 2.084400 
Lat. Cape Finisterre 42° b& K Mer. parts - 2858 

Mer. diff. - 121 
Lat. 41° 27' K, corresponding to - - - 2737 in table 

As cosine course 9.827085 

: proper diff. lat. 89 miles - - 1.949390 

: : radius 10.000000 

: dis. 132.5 miles 2.122305 

2. A ship from lat. 40° 41' K, Ion. 16° 37' W. y sails in the i\ r . 
E. quarter till she arrives in lat. 43° 57" A 7 !, and has made 248 miles 
departure ; required her course, distance, and longitude in. 

Ans. course K 51° 41' E., dis. 316 miles, and Ion. in 11° W. 

3. How far must a ship sail K E. -J E. from lat. 44° 12' JV., Ion. 
23° W., to reach the parallel of 47° A"., and what from that point 
will be the bearing and distance of Ushant, which is in lat. 48° 28' 
JV. and Ion. 5° 3' W. ? 

Ans. She must sail 262 miles, and her course and distance tc 
Ushant will then be K 80° 32' E., and dis. 535 miles. 

4. A ship from the Cape of Good Hope steers E. \ S. 446 miles, 
required her place, and her course, and distance to Kerguelen'.* 
Land, in lat. 48° 41' £., and Ion. 69° east. 



MECATOR'S SAILNIG. 197 

Ans. lat. in 35° 13' &, Ion. in 27° 21' E., course S. 66° 25' ^., 
and distance 2018 miles. 

5. By observation, a ship was found to be in lat. 41° 50' S., Ion. 
68° 14 E. She then sailed # E. 140m, and J' £ S. 76m; required 
her place, and her course, and distance to the island of St. Paul, 
which is in lat. 38° 42' 8., and in Ion. 77° 18' E. 

Ans. lat 40° 18' S., Ion. 72° 2', course N. 68° 35' E, and dis. 
263 miles, nearly. 



CELESTIAL OBSERVATIONS. 



CHAPTER I. 



We now come to the more scientific and essential parts of navi- 
gation, the determination of latitude and longitude by celestial 
observations. 

We shall at present confine ourselves to latitude, first calling to 
mind the following necessary definitions and explanations : 

1 . Meridian. — The meridian of any place is the north and south 
line passing through that place, and it may be conceived to run along 
the ground or pass in the same direction in the heavens, through the 
point vertically over the place. The line on the earth is the terres- 
trial meridian, the line in the heavens is called the celestial meridian ; 
they are both in one plane with the center of the earth. 

2. Equator. — The equator is that circle around the earth over 
which the sun seems to pass when the days and nights are equal 
all over the earth. 

3. Latitude. — The latitude of any place is the meridian distance 
of that place from the equator, measured by degrees and parts of a 
degree of arc. 

4. Longitude. — The longitude of any place is the inclination of 
the plane of its meridian, with the plane of some other definite 
meridian from which reckoning is made. This inclination is meas- 
ured on the equator by degrees, minutes, and seconds of arc, and it 
is either east or west.* 

* The first meridian to reckon from may be arbitrarily chosen, and different 
nations have taken different meridians for the commencement of longitude, but 
custom and long association have pretty firmly fixed the meridian of Greenwich 
(England) as the first meridian for all who use the English language. 
(198) 



NAVIGATION. 199 

5. Declination. — The declination of a heavenly body is its 
meridian distance from the equator north or south. 

6. Polar Distance. — The polar distance of a body is its decli- 
nation added to, or subtracted from 90°. If both added and sub- 
tracted, we shall have the meridian distances from each pole. 

The distance from the north pole, is called north polar distance, 
and from the south pole, south polar distance. The two polar dis- 
tances must of course make 1 80°. 

7. Zenith. — Zenith is the point in the heavens directly over- 
head. 

8. Horizon. — The horizon is either apparent or real, or as com- 
monly expressed, sensible or rational. 

The sensible horizon is a plane conceived to touch the earth at any 
point at which an observer is situated. 

The rational horizon is a plane parallel to the sensible one, passing 
through the center of the earth. 

The zenith is the pole to the horizon. 

9. Great Circles. — A great circle in the heavens is any circle 
whose plane passes through the center of the earth. 

All great circles which pass through the zenith are perpendicular 
to the horizon, and such circles are called vertical circles, azimuth 
circles, or circles of altitude. 

10. Azimuth. — The angle which the meridian makes with that 
vertical circle which passes through any object is said to be the 
azimuth of that object. Hence, azimuths may be reckoned from the 
north or south points of the horizon. 

11. Altitude. — The altitude of any object is its angular distance 
from the horizon, measured on a vertical circle.* 

Altitudes are very frequently measured at sea, several times in a 
day in fair weather ; but altitudes observed from the surface of the 
earth, or above it, require several corrections before the true alti- 
tudes can be deduced from them. 

* We do not pretend to give all the definitions of the sphere, but we suppose 
the reader is already acquainted with them, from his knowledge of Geography 
and Astronomy. 



200 



CELESTIAL OBSERVATIONS. 



These corrections are for semi-diameter, dip, refraction, 2lIl& parallax. 
The correction for semi-diameter is obvious. 

At sea, the visible horizon (from -which all observed altitudes are 
taken) is where the sea and sky apparently meet, and when the eye 
of the observer is above the water, this visible horizon is below the 
sensible horizon, and the amount of the depression is called the dip 
of the horizon. Its correction is always subtractive, and its amount 
is to be found in Table V. 

Refraction is to be found in Table VII. It is always subtractive, 
and for the reason, see some treatise on natural philosophy. 

Parallax is always additive. Conceive two lines drawn to a hea- 
venly body ; one from an observer at the circumference of the earth 
and the other from the center of the earth, the inclination of these 
two lines is parallax, and when the body is in the horizon its parallax 
is greatest, and it is then called horizontal parallax. 

Parallax always tends to depress the object, but the parallax of 
any celestial object, except that of the moon, is so small, that we 
shall pay attention to lunar parallax only, but this is so important to 
navigation that we shall give it a full explanation. 

The moon's horizontal parallax 
is given in the Nautical Almanac for 
every noon and midnight of Green- 
wich time, and from the horizontal 
parallax we must deduce the paral- 
lax corresponding to any other 
altitude. 

Let AC be the radius of the 
earth, A the position of an observer, 
Z his zenith, and suppose H to be the moon in the horizon ; then 
the angle A HO* is the moon's horizontal parallax, and the angle 
AhC is the parallax corresponding to the apparent altitude hAH. 
Draw Am parallel to Ch, then mAH would be the true altitude. 




*From this figure we draw the following definition for horizontal parallax. 

The horizontal parallax of any body is the angle under whichthe semi-diameter of 
the earth would appear as seen from that body. Of course then, when the body is 
at a great distance its horizontal parallax must be small, hence the sun and the 
remote, planets have very little parallax, and the fixed stars none at all. 



NAVIGATION. 201 

Let CH and Ch be each represented by R. Put j p= the hori- 
zontal parallax, and x= the parallax in altitude, or the angle mAh 
or Ah C. 

Now in the triangle A CH, right-angled at A, we have 

1 : sin.^> : : R : AC. 
In the triangle A Ch we have 

sin. CAh : sin. x : : R : AC. 
By comparing these two proportions, we perceive that 

1 : sin. p : : sin. CAh : sin. x 
Whence, sin. x= sin. p. sin. CAh 

But sin. CAh— cos. hAH, for the sine of any arc greater than 90° 
is equal to the cosine of the excess over 90°, hence, 
sin.#= sm.p cos. hAH 
The lunar horizontal parallax is rarely over a degree, commonly 
less, and the sine of a degree does not materially differ from the arc 
itself, hence, the preceding equation becomes the following, without 
any essential error. 

That is, x=p cos. altitude. 

Or, in words, the parallax in altitude is equal to the horizontal 
parallax multiplied into the cosine of the apparent altitude ( radius 
being unity). 

EXAMPLES. 

1. The apparent altitude of the moon's center after being corrected 
for dip and refraction was 31° 25'; and its horizontal parallax at 
that time, taken from a nautical almanac, was 57' 37"; what was the 
correction for parallax, and what was the true altitude as seen from 
the center of the earth ? 

p=5T 37 ,, =3457" log. - - 3.538699 
31° 25' cos. - - 9.931152 



z=49 / 10"=2950 log. - - 3.469851 

Ans. Cor. for parallax 49' 10" 
True altitude 32° 14' 10" 
2. The apparent altitude of the moon's center on a certain occa- 
sion was 42° 17'; and its horizontal parallax at the same time was 
58' 12"; what was the parallax in altitude, and what was the moon's 
true altitude? Ans. Parallax in alt. 43' 4" 

True alt. 43° 0' 4" 



202 



CELESTIAL OBSERVATIONS. 



No other examples of this kind are necessary, as they will inciden- 
tally occur in several places further on. 

It now remains to describe the instrument used for taking angles 
at sea. We, therefore, give the following illustrations on the 

QUADRANT AND SEXTANT. 




The quadrant and sextant are essentially the same instrument, 
and the following is an explanation of the principle on which they 
are constructed. 

Let AB be a section of a reflecting sur- 
face, FB a ray of light falling upon it, and 
reflected again in the direction BE, and BD 
a perpendicular at the point of impact ; then 
it is a well known optical fact, that the angles 
FB and EBA are equal, and that FB, DB, and FB are in the 
same plane. 

Again, if A were a reflecting surface, 
and a ray of light, SB, from any celestial 
object S, were reflected to an eye at F, the 
image of the object would appear at S' on 
the other side of the plane, the angles SB A 
and ABS', as well as FBC, being equal ; 
and if FB bear no sensible proportion to 
the distance of S, the angles SFS' and 
SBS' may be considered as equal ; for 
their difference, BSF, will be of no sensible magnitude. 

Before we proceed to the direct description of the sextant, it is 
necessary to give the following important 




LEMMA . 

If the exterior angle of a triangle he bisected, and also one of the 
interior opposite angles, and the bisecting lines produced until they 
meet, the angle so formed will be half the other interior opposite angle. 

Let AB C be the triangle, and bisect the exterior angle A CD 
by the line CF, and the angle Bbj the line BE. 

The angle E will be half the angle A. 

Let each of the angles ACE, ECD, be designated by x ( as rep- 




THE PLANE TABLE. 203 

resented in the figure ), and 
each of the equal parts of the 
angle B by y. Let A repre- 
sent the angle A, and E the 
angle E. 

Now as the sum of the three 
angles of every plane triangle 
is equal to 180°; therefore, in the the triangle ABO, we have 

-4+2y+C=180° (1) 

Also, in the triangle EB C, we have 

E-\-y+C+x=lS0° (2) 

Subtracting (2) from (1) gives us 

A—E+y—x=0 (3) 

Whence, A=E^(x—y) (4) 

But because # is the exterior angle of the triangle ECB 

x=E-\-y (see Elementry Geometry.) 

Or, (x— y)=E 

This value of (x — y) substituted in (4) gives 

A 
A=2E } or E=-£ Q. E. D. 

Another Demonstration.— The angle x being the half of A CD is 

equal to 

A+2y 

2 
The angle x is also equal to E-\-y, because it is the exterior angle 
to the triangle EBC. 
Therefore, by equality, 

2 
Whence, E=\A Q. E. D. 



204 



CELESTIAL OBSERVATIONS. 




We are now prepared to 
show the construction of 
the sextant and quadrant. 

The instrument repre- 
sented by the annexed cut 
is a quadrant or a sextant, 
according as the arc con- 
tains 90° or 120°, but each 
actual degree of arc is 
graduated to 2°, and the 
space that covers 90° is 
really but 45°, and so on. 

The reason why a half 
degree is counted and mark- 
ed as a whole one, we are 
about to explain. 

ABO is a firm plane sector, commonly made of metal or ebony; 
A J is a revolving index bar, turning on the center A, to which is 
attached a vernier scale, revolving over the graduated arc. 

The graduation commences at B. At A is a small plane mirror, 
perpendicular to the plane of the sector, it is attached to the index bar 
and revolves with it. This is called the index mirror or index glass. 

At H is another small mirror, half silvered and the other half 
transparent. This is called the horizon glass ; it might be called 
the image glass. 

The horizon glass must be perpendicular to the instrument, and 
parallel to AB. 

Now conceive a ray of light coming from an object S, striking the 
mirror A, the index and mirror being turned so as to throw the re- 
flecting ray into the mirror H, this mirror agains reflects it toward 
E, and an eye anywhere in the line Z>i/will see the image of the 
object behind the mirror H. Conceive the ray of light from S to 
pass right through the mirror at A, to meet the line HE) then, it 
is obvious that the angle SED measures the angle between the 
object S and its image D. 

Now, in the triangle A Elf, by a little inspection, it will be found 
that HL bisects the exterior angle, and A J, the index, bisects one of 



NAVIGATION. 205 

the interior opposite angles ; therefore, by the preceding lemma, the 
angle HLA is half the angle at E, but as AB and the mirror H are 
parallel, the angle HLA is equal JAB. It is obvious that JAB is 
measured by the arc BJ, or it measures the angle at E, if half de- 
grees on BO are counted as whole ones, which was to be shown. 

A tube, and sometimes a small telescope, is attached to the bar 
AB, and placed in the direction of the line Elf. This is called the 
line of sight. 

THE ADJUSTMENT OF THE INSTRUMENT. 

When this instrument is in adjustment, the two mirrors are per- 
pendicular to the plane of the sector, and are parallel to each other 
when on the vernier coincides with on the arc. We therefore 
inquire : First 

Is the index mirror perpendicular to the plane of the instrument? 

The following experiment decides the question. 

Put the index on about the middle of the arch, and look into the 
index mirror, and you will see part of the arch reflected, and the 
same part direct ; and if the arch appears perfect, the mirror is in 
adjustment ; but if the arch appears broken, the mirror is not in 
adjustment, and must be put so by a screw behind it, adapted to 
this purpose. Second, 

Are the mirrors parallel when the index is atO* 

Place the index at 0, and clamp it fast ; then look at some well- 
defined, distant object, like an even portion of the distant horizon, 
and see part of it in the mirror of the horizon glass, and the other 
part through the transparent part of the glass ; and, if the whole 
has a natural appearance, the same as without the instrument, the 
mirrors are parallel ; but, if the object appears broken and distorted, 
the mirrors are not parallel, and must be made so, by means of the 
lever and screws attached to the horizon glass. Third, 

Is the horizon glass perpendicular to the plane of the instrument? 

The former adjustments being made, place the index at 0, and 
clamp it ; look at some smooth line of the distant horizon, while 
holding the instrument perpendicular ; a continued unbroken line 
will be seen in both parts of the horizon glass ; and if, on turning 
the instrument from the perpendicular, the horizontal line continues 



206 CELESTIAL OBSERVATIONS. 

unbroken, the horizon glass is in full adjustment ; but, if a break in 
the line is observed, the glass is not perpendicular to the plane of the 
instrument, and must be made so, by the screw adapted to that 
purpose. 

After an instrument has been examined according to these direc- 
tions, it may be considered as in an approximate adjustment — a re- 
examination will render it more perfect — and, finally, we may find 
its index error as follows : — measure the sun's diameter both on and 
off the arch — that is, both ways from 0, and if it measures the 
same, there is no index error ; but if there is a difference, half that 
difference will be the index error, additive, if the greater measure is 
off the arch, subtractive, if on the arch. 

To measure the altitude of the sun at sea. 

Turn down the proper screen or screens, to defend the eye. Put 
the index at 0, having it loose, look directly at the sun through the 
tube, and you will see its image in the silvered part of the horizon 
glass. Now move the index, and the image will drop ; drop it to 
the horizon, and clamp the index. 

Let the instrument slightly vibrate each side of the perpendicular, 
on the line of sight as a center, and the image of the sun will appar- 
ently sweep along the horizon in a circle. While thus sweeping, 
move the tangent screw,* so that the lower limb of the sun will just 
touch the horizon, without going below it. The reading of the index 
will be the altitude corresponding to that instant, provided there be 
no index error. 

To measure the angular distance between two bodies as the sun and 
moon, or the moon and a star. 

The most brilliant of the two objects is always reflected to the 
other. Loosen the index, place it at 0, and direct the line of sight 
to the brighter object, and catch a view of its image in the silvered 
part of the horizon glass. 

Turn the plane of the instrument into the plane between the two 
objects ; now move the index, keeping the eye on the image, and- 

* The screens, adjusting screws, clamp screw, and tangent screw, are not given 
in our description of the instrument, it is not necessary to describe them; should 
we attempt it, there is danger that the spirit and clearness of the description 
would be lost in the multitude of words. 



NAVIGATION. 207 

bring it along to the other object ; bring them as near as possible, 
then gently clamp the index. 

Hold up the instrument again, in the plane between the two ob- 
jects, and view one object through the transparent part of the 
horizon glass ; and when the instrument is in the right position, the 
image of the other object will appear also in the same field of view, 
and then with the tangent screw, make the limb of the reflected 
object just touch the other, as it moves past it to and fro, by the 
gentle motion of the instrument. When the observer is satisfied 
that he has got the measure as near as he can, he cries out, 
mark, and his assistants mark the time by the watch, and the alti- 
tudes of the objects are also marked for the same time, if required, 
and observers are present to take them. 

The first experiments in the use of this instrument, other than 
measuring a simple altitude, are generally failures, but a little prac- 
tice will establish dexterity, skill, and confidence. 

We are now prepared to give examples for finding latitude. 

Let it be remembered, that latitude is the observer's zenith dis- 
tance from the equator, and the nautical almanac gives the distance 
of all the heavenly bodies from the equator, under the name of 
Declination. We can therefore observe our zenith distance from any 
celestial object, and then apply its declination, and we shall have 
our zenith distance from the equator, which is the latitude. 

EXAMPLES. 

1. On a certain day, the meridian* altitude of the sun's lower 
limb was observed to be 31° 44', bearing south. At that time its 
declination was 7° 25' 8" south, semi-diameter 16' 9", index error 
-f.2' 12", height of the eye 17 feet. What was the latitude ? 

Ans. 50° 38' north. 

* To obtain the meridian altitude of the sun, the observer commences obser- 
vations before noon, while the sun is still rising ; driving the index forward as 
fast as the image appears to rise, and there will come a time, a few minutes in 
succession, in which the image appears to rest on the horizon, neither rises nor 
falls, but at length the image will fall ; then the observer knows that noon has 
passed, and the greatest apparent altitude will be shown by reading the index. 



208 



NAVIGATION 



Semi-diameter -J- 16' 9" KA. 



Index error 

Refraction 

Dip 

Sum 



+ 



2.12 
1.31 
4.04 



Table, 
Table, 



-J-12'46' 



Alt. ob. 
Correction 
Alt. O 's center 

O's zenith dis. 
Dec. south 



31° 44 

12' 46" 



31 


56 


46 


90° 






58° 


3' 


14" 


7° 25' 


8" 



Latitude north 50° %8' 6" 

In this example, if the meridian altitude had been observed in the 
north, in place of the south, what then would have been the ob- 
server's latitude ? Ans. 65° 28' 22" south. 

We may note the following 

Rule. — Subtract the corrected altitude from 90°. Then if the 
observer and the object are both on the same side of the equator, add the 
declination, but if on different sides, subtract the declination, and the 
sum or difference will be the latitude of the observer. 

Find the latitude from each of the folio win £ meridian observations: 



Object. 

" L.L. 

Jupiter 

Saturn 

Sirius 

Sun U. L. 

Sun L. L, 

Sun L. L, 



Alt. ob, 



45o 27' 
81° 43' 
73o 17' 
82° 12' 
750 5' 
40O42 
87Q29' 
150 45 



Direc. 



South 
South 
South 
North 
North 
North 
South 
South 



S.D. 
16' 15' 
15' 47' 



16' 17' 
16' 17' 
16' 0' 



Heigbt. 
207eet 
14 " 
17 « 

17 « 

18 « 
16 " 
16 " 
16 " 



Declination. 



17° 19' 31" S. 
22° 13' T'N. 
24° 10' 13" S. 
12° 9' Q"N. 
16° 31' S. 

23022' S. 

220 9' S. 
4°43' S. 



Latitude. 



27o 2'51"iV. 

30o 18' 10" N. 
7o 22' 52" S 
40 16' 54" N. 

310 30'26" & 

73° 1' 19" S. 

19o 50' 12" S. 

690 23'16"iV 



In this table L. L. indicates lower limb, U. L. upper limb, S. D. semi 
diameter, N. north, <S. south, Direc. direction. In these examples, the instru 
ment is supposed to have no index error. 

Night observations at sea are of little value, for it is very seldom 
that the horizon can be defined, unless it is in bright moon-light, in 
the tropical climates. 

For this reason, very few navigators attempt to find the latitude, 
by observations on the planets and fixed stars. 

Occasionally, however, when one of the bright planets, or a con- 
spicuous fixed star, comes to the meridian in the morning or evening, 
twilight observations can be made on them, and the latitude deduced. 

Some navigators apply a summary correction to the sun's lower 
limb, for semi-diameter, dip, and refraction, such as is comprised in 
the following table. 



CELESTIAL OBSERVATIONS. 



209 



l> = 


Correction to be added to the 


Observed 


Altitude 


of the 




Sfon's Lower Limb, to find the True Altitude 






Height of the Eye ahove the Sea in Feet. 


6 


8 


10 


12 


14 , 16 


18 


20 


22 


24 


26 


28 


30 


32 


34 


~o~ 


/ 


— ', 


/ 


' 


~~i~ 


' 


' 


' 


T~ 


~~ '" 


i 


' 


~ r 




/ 


5 


3.8 


3.5 


3.1 


2.8 


2.5 


2.3 


2.1 


1.8 


1.6 


1.4 


1.2 


1.0 


0.8 


0.6 


0.5 


6 


5.3 


4.9 


4.6 


4.3 


4.0 


3.7 


3.5 


3.3 


3.0 


2.8 


2.6 


2.4 


2.2 


2.1 


1.9 


7 


6.4 


6.0 


5.7 


5.4 


5.1 


4.8 


4.6 


4.4 


4.1 


3.9 


3.7 


3.5 


3.3 


3.2 


3.0 


8 


7.2 


6.8 


6.5 


6.2 


5.9 


5.7 


5.4 


5.3 


5.0 


4.8 


4.6 


4.4 


4.2 


4.0 


3.9 


9 


7.9 


7.5 


7.2 


6.9 


6.6 


6.4 


6.1 


5.9 


5.7 


5.5 


5.3 


5.1 


4.9 


4.7 


4.5 


10 


8.5 


8.1 


7.8 


7.5 


7.2 


6.9 


6.7 


6.5 


6.2 


6.0 


5.8 


5.6 


5.4 


5.3 


5.1 


11 


8.9 


8.6 


8.2 


7.9 


7.6 


7.4 


7.2 


6.9 


6.7 


6.5 


6.3 


6.1 


5.9 


5.7 


5.6 


12 


9.3 


9.0 


8.7 


8.3 


8.0 


7.8 


7.6 


7.3 


7.1 


6.9 


6.7 


6.5 


6.3 


6.2 


6.0 


14 


9.9 


9.6 


9.2 


8.9 


8.7 


8.4 


8.2 


7.9 


7.7 


7.5 


7.3 


7.1 


6.9 


6.8 


6.6 


16 


10.4 


10.1 


9.7 


9.4 


9.1 


8.9 


8.7 


8.4 


8.2 


8.0 


7.8 


7.0 


7.4 


7.2 


.7.1 


18 


10.8 


10.4 


10.1 


9.8 


9.5 


9.3 


9.0 


8.8 


8.6 


8.4 


8.2 


8.0 


7.8 


7.6 


7.5 


20 


11.1 


10.7 


10.4 


10.1 


9.8 


9.6 


9.3 


9.1 


8.9 


8.7 


8.5 


8.2 


8.1 


7.9 


7.7 


22 


11.4 


11.0 


10.7 


10.4 


10.1 


98 


9.6 


9.4 


9.1 


8.9 


8.7 


8.5 


8.3 


8.2 


8.0 


26 


11.7 


11.4 


11.0 


10.7 


10.5 


10.2 


10.0 


9.7 


9.5 


9.3 


9.1 


8.9 


8.7 


8.6 


8.4 


30 


12.0 


11.7 


11.3 


11.0 


10.8 


10.5 


103 


10.0 


9.8 


9.6 


9.4 


9.2 


9.0 


8.9 


8.7 


35 


12.3 


11.9 


11.6 


11.3 


11.0 


10.7 


10.6 


10.3 


10.1 


9.9 


9.7 


9.4 


9.2 


9.2 


9.0 


40 


12.5 


12.2 


11.8 


11.5 


11.3 


11.0 


10.8 


10.5 


10.3 


10.1 


9.9 


9.7 


9.5 


9.4 


9.2 


45 


12.7 


12.4 


12.0 


11.7 


11.5 


11.2 


11.0 


10.7 


10.5 


10.2 


10.1 


9.8 


9.7 


9.6 


9.4 


50 


12.8 


12.5 


12.2 


11.9 


11.6 


11.3 


11.1 


10.9 


10.6 


10.4 


10.3 


100 


9.8 


9.7 


9.5 


55 


13.0 


12.6 


12.3 


12.0 


11.7 


11.5 


11.2 


11.0 


10.7 


10.5 


10.3 


10.1 


9.9 


9.8 


9.6 


60 


13.1 


12.7 


12.4 


12.1 


11.8 


11.6 


11.3 


11.1 


10.9 


10.6 


10.4 


10.2 


10.1 


9.9 


9.7 


65 


13.2 


12.8 


12.5 


12.2 


11.9 


11.7 


11.4 


11.2 


11.0 


10.7 


10.5 


10.3 


10.1 


10.0 


9.8 


70 


13.3 


12.9 


12.6 


12.3 


12.0 


11.8 


11.5 


11.3 


11.0 


10.8 


10.6 


10.4 


10.2 


10.1 


9.9 


75 


13.4 


13.1 


12.7 


12.4 


12.1 


11.9 


11.7 


11.4 


11.2 


11.0 


10.8 


10.6 


10.4 


10.2 


10.1 


80 


13.6 


13.2 


12.9 


12.6 


12.3 


12.0 


11.8 


11.6 


11.3 


11.1 


10.9 


10.7 


10.5 


10.4 


10.2 


Monthly 


Jan. 


Feb. 


Mar. 


April, 




Ma 3 




June, 


Correction 
for Sun's 


-l-0'3 


-l-0'.2 


-fO'.l 


O'O 




— 0' 


i 


— 0'.2 


July.. 


Aug. 


Sept. 


Oct. 




fcoi 




Dec. 


Semi-diam. 


— 0'.3 


— 0'.2 


— O'.l 


+0U 




4-0' 


.2 


-f0'.3 



The most practical method of obtaining the latitude by observa- 
tion, other than the meridian altitude of the sun, is by the meridian 
altitude of the moon ; but to correct the observed altitude for 
semi-diameter, parallax, refraction, and dip, and do it to the utmost 
accuracy, requires more computation and attention than the mere 
practical navigator is disposed to give. Moreover, such like accuracy 
is not required in practical navigation. To know the latitude within a 
mile is all the ship master requires ; and this can be done in a very 
summary manner, by observing the moon's meridian alititude and 
using, the following tables, according as the lower* or upper limb 
of the moon is observed. 



* The bright limb, is the one observed, whether it be the upper or lower. 
14 



210 



NAVIGATION. 



These tables make but one correction for semi-diameter, parallax and refraction 











TABLE I. 












Corrections to be added to the Observed Altitude of the Moon 


's lower limb. 


Part 1st. HORIZONTAL PARALLAX. 


d's 
Alt 


53' 

o / 


54' 
o / 


55' 
o / 


56' 


57' 


58' 


59' 


60' 


61' 


o / 


o / 


o / 


o 7 


o T~ 


"5 T 


~6 


0.59 


1. 


1. 1 


1. 3 


1. 4 


1. 5 


1. 6 


1. 8 


1. 9 


8 


1. 


1. 2 


1. 3 


1. 4 


1. 6 


1. 7 


1. 8 


1. 9 


1.11 


10 


1. 1 


1. 3 


1. 4 


1. 5 


1. 7 


1. 8 


1. 9 


1.10 


1.12 


15 


1. 2 


1. 3 


1. 5 


1. 6 


1. 7 


1. 9 


1.10 


1.11 


1.12 


20 


1. 2 


1. 3 


1. 4 


1. 5 


1. 6 


1. 8 


1. 9 


1.10 


1.11 


25 


1. 


1. 2 


1. 3 


1. 4 


1. 5 


1. 6 


1. 7 


1. 9 


1.10 


30 


0.59 


1. 


1. 1 


1. 2 


1. 3 


1. 4 


1. 5 


1. 7 


1. 8 


35 


0.57 


0.59 


0.59 


1. 


1. 1 


1. 2 


1. 3 


1. 4 


1. 5 


40 


0.55 


0.55 


0.56 


0.57 


0.58 


0.59 


1. 


1. 1 


1. 2 


45 


0.51 


0.52 


0.53 


0.54 


0.55 


0.56 


0.57 


0.58 


0.59 


50 


0.48 


0.49 


0.50 


0.51 


0.51 


0.52 


0.53 


0.54 


0.55 


55 


0.44 


0.45 


0.46 


0.47 


0.48 


0.49 


0.49 


0.50 


0.51 


60 


0.40 


0.41 


0.42 


0.43 


0.44 


0.44 


0.45 


0.46 


0.47 


65 


0.36 


0.37 


0.38 


0.39 


0.39 


0.40 


0.40 


0.41 


0.42 


70 


0.33 


0.33 


0.34 


0.34 


0.35 


0.36 


0.36 


0.37 


0.37 


75 


0.28 


0.28 


0.29 


0.29 


0.30 


0.30 


0.31 


0.31 


0.32 


60 


0.24 


0.24 


0.24 


0.25 


0.25 


0.25 


26 


0.26 


0.27 


85 


0.19 


0.19 


0.19 


0.20 


0.20 


0.21 


0.21 


0.21 


0.22 


TAB 


LEII. C 


ORRECTIOl 


fs to be a 


pplied to 


Observei 


> Altitud 


e of the B 


loon's up 


per linib 








Part 


2nd. hor 


izontal 1 


'ARALLAX 






d's 


53' 


54' 


55' 


56' 


57' 


58' 


59' 


60' 


61' 


Alt. 


o / 


o / 


O 1 


o / 


o / 


o / 


o / 


o / 


o / 


To 


■4-0.33 


4-0.33 


4-0.34 


4-0.34 


4-0.34 


4-0.36 


4-0.37 


4-0.37 


+ 0.38 


15 


0.33 


0.33 


0.35 


0.35 


0.36 


0.37 


0.37 


0.39 


0.39 


20 


5.32 


0.33 


0.34 


0.35 


0.35 


0.36 


0.37 


0.37 


0.38 


26 


0.30 


0.32 


0.32 


0.33 


0.33 


0.34 


0.35 


0.35 


0.36 


30 


0.29 


0.30 


0.31 


0.31 


0.32 


0.32 


0.33 


0.34 


0.34 


36 


0.26 


0.26 


0.27 


0.27 


0.28 


0.28 


0.29 


0.29 


0.30 


40 


0.24 


0.25 


0.26 


0.26 


0.27 


0.27 


0.28 


0.29 


0.29 


46 


0.19 


0.22 


0.22 


0.22 


0.23 


0.24 


0.24 


0.24 


0.26 


50 


0.17 


0.19 


0.19 


0.20 


0.20 


0.21 


0.21 


0.21 


0.21 


55 


0.14 


0.15 


0.16 


0.16 


0.16 


0.16 


0.17 


0.17 


0.17 


60 


0.10 


0.11 


0.12 


0.12 


0.12 


0.12 


0.13 


0.13 


0.13 


65 


0. 6 


0. 7 


0. 7 


0. 8 


0. 6 


0. 8 


0. 8 


0. 8 


0. 9 


70 


0. 3 


0. 3 


0. 3 


0. 3 


0. 3 


0. 3 


0. 3 


0. 3 


0. 4 


75 


— 0. 1 


— 0. 1 


— 0. 1 


— 0. 1 


— 0. 1 


— 0. 2 


— 0. 2 


— 0. 2 


— 0. 2 


80 


0. 6 


0. 6 


0. 6 


0. 6 


0. 6 


0. 6 


— 0. 6 


0. 7 


0. 7 


85 


0.10 


0.11 


0.11 


0.11 


0.11 


0.11 


0.11 


0.12 


0.12 


Heig 
Dipc 


it of the 
f the Hoi 


eye, 






4ft. | 


9ft. | 


16ft. | 


25ft. | 


36ft. 


•izon, 






-2' | 


—-b<~\ 


~—& r 



EXAMPLES. 



1. In longitude about 45° west, on the 5th of January, 1852, at 
about 1 lh. in the evening, I observed the altitude of the moon's 
lower limb as she passed the meridian, and found it to be 68° 12' 
from the south, height of the eye 16 feet. What was my latitude ? 



CELESTIAL OBSERVATIONS. 211 

On the 5th of Jan., at llh. evening, Ion. 45 west, corresponds to 2 hours 
after midnight at Greenwich. 

From the Nautical Almanac, we find, that, 

At midnight of the 5th, the moon's horizontal parallax was - - 57' 40 A 
At noon of the 6th, 58' 1' 

Therefore, by proportion, the horizontal parallax at the time of observation, 
must have been 57' 43". 

Moon's declination at midnight of the 5th, (N. Almanac), 21° 47' 53" N. 
" " noon of the 6th, ... - 22° 16' 55" JV 



Variation in 12 hours, ---___- 29' 2" 

Therefore, the variation for 2 hours, was not far from - 4' 50" 

Hence the dec. at the time of observation was, - - 22° 52' 43" A r . 

We enter table t, and under the parallax, and opposite to the 
altitude as near as we can find them, we perceive that 37' must be 
about the correction for the altitude. 

Whence, Observed alt. L. L 
Correction, 

Dip. always sub. 



68° 


12' 


+ 


37 


68 


49 


— 


4 


68 


45 


90 




21 


15 


22 


53 



Zenith dis. } 
J) 's dec. 

Lat. in 44° 8' North. 

Find the true altitude of the moon's center, in each of the follow- 
ing examples. L. L. means lower limb ; U. L. upper limb. 





Observed Alt. 


H. P. 


Height of 
the eye. 


Ans. 
True Alt. 


1. 


D L.L. 


53° 23' 


58' 14" 


14 feet 


54° 10' nearly 


2. 


D L.L. 


48° 58 


60' 27" 


19 " 


49° 48' « 


3. 


D U.L. 


57° 11' 


54' 30" 


20 " 


57° 19' 't 


4. 


D L.L. 


63° 38' 


55' 29" 


12 " 


64° 14' " 


5. 


D U.L. 


20° 3' 


54' 14" 


16 " 


20° 32' " 


6. 


D L.L. 


16° 2' 


59' 38" 


23 " 


17° 12' " 



When the weather makes it doubtful whether meridian observa- 
tions can be obtained, navigator's resort to double altitudes, or to the 
altitudes of two objects taken at the same time. We shall only show 



212 



NAVIGATION. 




the principle on which this method is founded ; it is the application 
of spherical trigonometry. 

Let Ph be the earth's axis, Qq 
the equator. Suppose the sun to be 
the object, and let its position be S 
and T at two different times. 

The elapsed time measures the 
angle SPT. In the triangle PTS, 
we have the two sides PT, PS, and 
the included angle, from which we 
compute the side TS, and the angle 
TSP. 

Subtracting the altitudes Sm and Tn from 90°, we have ZS, and 
ZT, then we have all the sides of the triangle ZTS, from which we 
compute the angle TSZ. Subtracting this angle from TSP, gives 
us the angle ZSP. Now, in the triangle ZSP, we have the two 
sides ZS, SP, and their included angle, from which we compute PZ 
the complement of the latitude. 

If the ship sails, during the interval between the observations, a 
correction will be required for the first altitude, and such corrections 
are found by the traverse table ; a nautical mile in the direction of 
the sun, corresponds to one minute of a degree, to be applied to the 
altitude. When the proper correction is made, the result is equiva- 
lent to having both altitudes taken at the last station, and the deduced 
latitude is the latitude of that station. 



CHAPTER II. 

LONGITUDE. 



Longitude, from celestial observations, is measured by time. A 
place 15° west of another, will have noon one hour of absolute 
time later ; if 30° west, the local time, noon will be two hours later, 
<fec, &c; 15° corresponding to an hour in time. Therefore, if we 
have any way of determining the times at two places, correspond- 
ing to the same absolute instant, the difference of such times will 



LONGITUDE. 213 

correspond to the difference of longitude between the two places at 
the rate of 15° to an hour, or 4 minutes to a degree. 

A perfect time piece will keep the time at any 'particular meridian, 
and by carrying that perfect time piece with us, by it we can see the 
time at that particular meridian ; and then if we can find the time at 
the place where we are, the comparison of these two times will give the 
difference of longitude, that is, the difference between our longitude 
and that of the particular meridian, to which the time piece refers. 

For instance, a gentleman leaves Boston ; his watch is a perfect 
time piece, and it is set to Boston time, he travels west on the rail- 
roads, his watch all the while shows Boston time ; when it is twelve 
o'clock by his watch it is really so in Boston, but not so at the place 
where he is. The sun has arrived at the meridian of Boston, 
but not yet at the meridian of Albany, or Buffalo, or Detroit ; and 
when the gentleman arrives at gay of these places, or any interme- 
diate place, the local time, compared with the time in Boston, will 
give the longitude of that locality from Boston, counting one degree 
for every four minutes in the difference of time. 

Unfortunately, however, there is no such thing as a perfect time 
piece, but some do approximate toward perfection. Such ones, made 
with the greatest care and solely for accuracy in rate of motion, are 
called chronometers ; they are supposed to keep time within certain 
known limits, and in the place of perfect time keepers, they are used 
at sea for finding longitude. 

Chronometers show the time at the distant place, it then remains 
to find the time at ship, and this is done most accurately by spheri- 
cal trigonometry, as will soon appear. 

The sun's altitude is greatest just at apparent noon, but by obser- 
vations we cannot define just the moment when that takes place ; 
hence meridian observations, valuable as they are for latitudes, are 
worth nothing for time, when time is to be settled to anything like 
accuracy. 

The best position of the sun ( or any other celestial object ) 
for an observation to find local time, is when it is nearly east or west, 
and its altitude more than ten decrees. 

In such circumstances, an observer can find the local time 



214 NAVIGATION. 

within 5 or 6 seconds, by taking an altitude of the sun, provided he 
at the same time knows his latitude and the sun's polar distance. 

The operation is a beautiful application of spherical trignometry, 
and it is illustrated by the following figure. 

Let Z be the zenith of the 
observer, P the pole, S the 
position of the sun, and PS 
the sun's polar distance.* 

When S comes on to the 
meridian, it is then apparent 
noon ; and the angle ZPS of 
the triangle ZPS measures 
the interval from apparent 
noon, at the rate of four min- 
utes to one degree. 

The side PS is the polar 
distance, the side ZS is the co-altitude, and the side PZ is the co- 
latitude. 

Now, in every treatise on spherical trigonometry, it is demon- 
strated as a fundamental principle, that 

The cosine of any angle, of a spherical triangle, is equal to the co- 
sine of its opposite side, diminished by the rectangle of the cosines of the 
adjacent sides, divided by the rectangle of the sines of the adjacent 
sides. 

r . cos. ZS— cos. PZ cos. PS 

1 hat is, cos. .7^= = n „ . — r>a 

sin. PZ sin. PS 

Now, in place of cos. ZS, we take its equal, sin. ST, or the sine 

of the altitude, and in place of cos. PZ, we take its equal, the sine 

of the latitude. 

In short, let .4= the altitude, L= the latitude, and D= the 
polar distance. 

sin. A — sin. L cos. D 




Then cos. P= 



cos. L sin. D 



* When the observer is in the northern hemisphere, the polar distance is 
counted from the north pole; when in the southern hemisphere, from the south 
Dole. 



LONGITUDE. 215 

From a general equation, in plane trigonometry, we have 

2 sin.2 %P=\—cos. P 
Substituting the value of cos. P, in this last equation, we have 

sin. A — sin. L cos. D 



sin. 2 iP=l- 



cos. L sin. D 



(cos. L sin. D-f-sin. L cos D) — sin. A 
cos. L sin. D 

By comparing the quantity in parentheses with eq. (7), plane 
trigonometry, we perceive that 

sin. (Z-f-Z>)— sin. A 



2 sin.2 i P= 



cos. L sin. i> 



Considering (L-\-D) to be a single arc, and then applying 

equation (16), plane trigonometry, and dividing by 2, we shall 
have 

/L+D-\-A\ . /L+D—A 

COS. 






\ . ( L-YD— A \ 



sin 2 .|P=_ 

cos. L sin D. 

But L + J) — A _ ^+^+^ _^ } an d now if we put 

2 ~~ 2 

#= o we shall have 

_ cos. aS sin. (S — ^4) 

sin 2 . X P= T ' n ' 

2 cos. L sm. D 



~ • 1 d /cos. £ sin. ( S — A ) 

Or, sm. | <P=a/ p4 — ^ 

v cos. i> sm. jt> 

This is the final result when radius is unity, when it is R times 
greater, then the sin. -J P will be R times greater, and if R repre- 
sents the radius of our tables, to correspond with these tables we 
must multiply the second member by R, and if we put it under the 
radical sign, we must multiply by R 2 ; in short we shall have, 

Sin. i P= S /(^J) (-JL?) cos. S sin. (S-A) 
^ \cos. LJ \sm. DJ 

The right hand member of this equation, shows four distinct 

r> 

logarithms ; thus, is the cosine of the latitude subtracted from 

cos. L 

10, which we shall call cosine complement. 



216 NAVIGATION. 

This equation furnishes the following rule for finding apparent 
local time, when the sun's altitude, its polar distance, and the lati- 
tude of the observer, are given. 

The altitude must be observed, the latitude must be known, and the 
Nautical Almanac will furnish the polar distance. 

Rule. — 1. Add together the altitude, latitude, and polar distance; take 
the half sum, and from the said half sum subtract the altitude, thus 
finding the remainder. 

2. The logarithms. Find the cosine complement of the latitude, the 
sine complement of the polar distance, the cosine of the half sum, and 
the sine of the remainder. 

3. Add these four logarithms together, and divide by 2, the logarithm 
thus found, is the sine of half the polar angle, or half the sun's 
meridian distance. 

4. Take out the arc corresponding to this sine, and divide its double 
by 15 (as in compound division in arithmetic), and the quotient will 
be the hours, minutes, and seconds from apparent noon ; and if the sun 
is east of the meridian, the hours, minutes, and seconds, must be sub- 
tracted from 1 2 hours, for the corresponding time of day. 

The time shown by a chronometer or a perfect clock, or rather 
graduation of clocks, is to mean and not to apparent time, and to 
convert apparent into mean time, the equation* of time is given in 
the Nautical Almanac for the noon of every day at Greenwich. The 
amount of it, reduced or modified to correspond to the time of obser- 
vation, can be applied to apparent time, and the mean time of taking 
the observation will be determined. The difference between this 
time and the mean time at Greenwich, as determined by the chro- 
nometer, will be the longitude. The longitude will be west, if the 
time at Greenwich is latest in the day, otherwise it will be east. 

If the observer is on land, without a sea horizon, and uses a 
reflecting instrument, he must have an artificial horizon. A proper 
artificial horizon, is a small dish of mercury, with a glass roof to put 
over it, to keep the mercury from being agitated by the wind. In 
place of the mercury, a plate of molasses will answer. In still calm 
weather any clear pool of water is a good artificial horizon. 

In either of these, the reflected image of the object appears as 
much below the horizon as it is above it, and to measure the altitude, 

* For the theory of equation of time, see works on astronomy. 



LONGITUDE. 217 

the image reflected by the mirror of the instrument must be carried 
to the image in the artificial horizon ; half of the angle shown by 
the index will be the apparent altitude. In using an artificial hori- 
zon there is no dip, other corrections are to be applied according to 
circumstances. 

EXAMPLES UNDER THE PRECEDING RULE. 

1. Being at sea, May 20th, 1823, in latitude 43° 30' i\ r ., and in 
longitude about 20° west, I observed the altitude of the sun's 
lower limb, and found it to be 32° 4' rising, when an assistant 
marked the time per watch, at 7h. 43m. A. M. ; height of the eye 
16 feet. What was the true mean time ? 

Just before the observation, the watch was compared with the 
chronometer in the cabin*, and found to be 1 hour, 21 minutes, and 
12 seconds slow of chronometer. 

On the 8th of May, the chronometer was 3m. 7s. fast of Greenwich 
time, and gaining Is. 6 daily. What was the longitude ? 
QS.D., - - - 4- 15' 49" 
Dip., - - - - — 3 56 
Ref., - - - - — 1 30 



Correction, - - - -f 10 23 
Observation, - 32 4 



Alt. O center, - - 32° 14' 23' 



H. M. S' 

Watch, - - - - 7 43 
Diff., - - - - - 1 21 12 



Face of chron. at ob., - 9 4 12 

Error 3m. 7s., increase of error 

in 12 days 19s., whole error, — 3 26 



Greenwich time, - - 9 46 

At noon on the 20th of May, 1823, the sun's declination, by the N. A., was 

19° 52' 18" north, increasing at the rate of 30".6 per hour, and the time of 

taking the observation was 3 hours before noon at Greenwich ; therefore, the 

declination must have been 19° 50' 47" N. 



Altitude, 


32° 14' 23" 






Lat., 


43 30 


cos. com. 


.139435 


P.D., 


70 9 13 


sin. com. 


.026603 



2)145 53 36 



S. 72 56 48 cosino 9.467253 

32 14 23 



(S—A) 40 42 25 sine 9.814363 

2)19.447657 
31° 58' 8" sin. 9.723828 



* Chronometers should never be, and by careful persons, never are, taken out 
of their places during a voyage. 



218 NAVIGATION 



31° 58' 8" 
2 


= 4h. 
12 


15m. 




63° 56' 16"= 


45s. 


Apparent time, - 
Equation of time N. A. 


7 


44 
3 


15 A. M. 

51 


Mean time at ship, 
Watch, - 


7 
- 7 

9A 

7 


40 
43 


24 


Watch too fast, - 

Time at Greenwich per chron., 

Time at ship per observation, 


2m 
. 0m 
40 


36s. 
46s. 
24 


Diff., - - 


1 


20 


22=20° 5' west Ion. 



2. August 10th, 1824, in latitude 54° 12' north, at 5h 33m per 
watch, height of the eye 18 feet, I observed the altitude of the 
sun's upper limb 16° 50' falling. My chronometer was 2h 20m 
37s fast of the watch ; and on the 7th, the same month, the chro- 
nometer was 40m 29.4s fast of Greenwich time, gaining 7-f^ seconds 
daily. What was the error of the watch, and the longitude per 
chronometer ? 

Preparation. 

Time per watch - 5h. 33m. 0s. P. M. 

Diff. per chron. - - 2 20 37 

Face of chronometer - 7 53 37 P. M. 
Chron. fast (whole error) — 40 52 

Greenwich, mean time, 7 12 45 P. M. 
On the 10 of August, 1824, the sun's declination at noon, Greenwich time, 
was 15° 32' 14" north, decreasing at the rate of 45" per hour, as given in the 
Nautical Almanac. The decrease for 1\ hours must be 5' 24"; whence, the 
declination at the time of observation, 15° 26' 50" JV., and the polar distance 
74° 33' 10". 



Observed altitude - 16° 50' 00" 

Semi-diameter, N. A. - — 15 48 

DipandRef. - - —7 20 

True alt. center - - 16° 26' 52" 



Equation of time, per N. A., Aug. 
10, 1824, was - - -f 5m. 2s. 
Hourly decrease T 3 ff 8 -s. — 2 

Equation at ob. - - 5m. 0s. 



We now leave the problem to be worked through by the pupil, giving only 
the answer. 

Ans. Watch slow of local mean time, 3m 27s. 
Longitude by chronometer, 24° 4' 30" west. 
3. When it was 6h Om 21s, P. M., mean time, at Greenwich, by 
my chronometer, I observed the altitude of the sun's lower limb to 
be 30° 17', in the afternoon of January 12th, 1852. At noon our 



LONGITUDE. 219 

latitude, by a meridian observation, was 21° 47' north, and since 
that time we have made 11 miles of southing, by the log. The 
dip was 4', and semi-diameter 16' 17". What was the longitude by 
chronometer ? 

Sun's Declination Jan. 12, '52, at noon, G. T. - 21° 44' 10" south. 
Hourly decrease, per N. A., 25", giving - — 2' 30" 

Declination at the time of observation - - 21° 41" 40 ' south 
Equation of time at noon, Greenwich - - -{-8m. 25s. 

Hourly increase ^ Q 6 S of a second, making - - 6s. nearly 

Equation at time of observation (to add) - -j-8m. 31s. 

Were we sure that pupils would have access to nautical almanacs, we would 
give neither declination nor equation of time. 

Ans. Lon. 46° 9' west. 

4. On the 16th of January, 1852, when my chronometer showed 
llh 27m 41s, A. M., for the mean time at Greenwich, I observed the 
altitude of the sun's lower limb and found it 32° 21' rising, height 
of the eye 16 feet, latitude 0° 41' south. What was the longitude 
by chronometer ? 

By the N. A., the sun's declination at that time was 21° 2' 36'' south, and 
the equation of time 9m. 53s. additive. 

Ans. Lon. 44° 39' west, 
N. B. — Time at any place, is but the difference between the 
right ascension of the meridian and the right ascension of the sun ; 
and to find the time from these two elements, we always subtract 
the right ascension of the sun from the right ascension of the meri- 
dian, increasing the latter by 24 hours, to render subtraction possi- 
ble, when necessary. 

The right ascensions of the stars are given, and the right ascen- 
sion of the sun is given, in the Nautical Almanac, for the noon of 
every day in the year, Greenwich time. Now, if we can find the 
meridian distance of any known star, by observation, we can estab- 
lish the right ascension of the meridian, and, consequently, the local 
time. Hence, we can find longitude by comparing the chronometer 
with the altitudes of the stars, as the following example will illus- 
trate. 

5. If on the 8th of March, 1852, when my chronometer showed the 
Greenwich time to be 7h 22m 3s, P. M., I found by observation, 



220 NAVIGATION 

that the true altitude of Sirius was 37° 52' west of the meridian. 
My latitude was 32° 28' south. What was the time at ship, and 
what was my longitude ; the elements for computation being as 
follows ? 

1. Right ascension of the star - - 6k. 38m. 38s. 

2. Declination of the star 16° 31' south 

3. Right ascension of the sun - r 23/*. 1 7m. 25s. 
By means of the triangle we find, 

The meridian distance of the star - 3k. 40m. 58s. 
To which add -fc's R. A., because -fc is west 6 38 38 

Right ascension of the meridian 

Add 

Subtract the R. A. of the sun 
Diff. is apparent time at ship 
Equation of time, add - 

Mean time at ship - 
Time at Greenwich - 

Longitude in time - - - 3 50 56 

= 57° 44' east 
N. B. — When the chronometer remains in the same place for a 
week or more, its rate can be determined by comparing it with the 
observed altitudes of the sun, taken from day to day. In different 
climates the same chronometer will have different rates, and on re- 
turning to its original station it will frequently resume its original 
rate. 

For azimuths, and variations of the compass, see page 106. 



10 


19 


36 


24 






34 


19 


36 


23 


17 


25 


11 


2 


11 P.M 




10 


48 


11 


12 


59 


7 


22 


3 



CHAPTER III. 

LUNAR OBSERVATIONS. 

A good and well-tried chronometer is a valuable and reliable 
instrument for finding the longitude at sea, during short runs ; but 
still it is but an instrument, and is not one of the reliable works of 



LUNAR OBSERVATIONS. 221 

nature. Near the end of a long voyage, the best of chronometers 
very frequently give false longitude, and in such cases, good navi- 
gators always resort to lunar observations, which from the hands of 
a good observer, can be relied upon to within 10 or 12 minutes of a 
degree, and they usually come within 5 or 6 miles, and sometimes 
even more exact, but that is accidental and unfrequent. 

To comprehend the theory of lunars, we must call to mind the fact 
that the moon moves through the heavens, apparently among the 
stars, at the rate of more than 13° in a day, and any angular dis- 
tance it may have from the sun or any star corresponds to some 
moment of Greenwich time. 

About three- days before and after the change of the moon, she is 
too near the sun to be visible, but at all other times, her distance 
from the sun, some of the larger planets, and certain bright fixed 
stars, called lunar stars,* which lie near her path, are computed and 
put down in the nautical almanac, for every third hour of mean 
Greenwich time commencing at noon. For any particular day, the 
distances are given to such objects only, east and west of her, as 
will be convenient to measure with the common instruments. 

The distances put down in the nautical almanac, are such as would 
be seen if viewed from the center of the earth ; but observers are 
always on the surface of the earth, and the distances thence observed, 
must always be reduced to equivalent distances seen from the center, 
and this reduction is called working a lunar, which is generally the 
highest scientific ambition of the young navigator. \ 

The true distance between the sun and moon, or between a star 
and the moon, can be deduced from the apparent distance by the 
application of spherical trigonometry. 

The moon is never seen by an observer in its true place, unless the 
observer is in a line between the center of the earth and the moon, 
that is, unless the moon is in the zenith of the observer ; in all other 

* There are nine lunar stars, Arietis, Aldebaran, Pollax, Regulus, Spica, 
Antares, Aquilae, Fomalhaut, and Pegasi. 

t Many navigators, both old and young, direct all their efforts to knowing how 
to do, without attempting to comprehend the reasons for so doing ; and this the 
world calls practical, — a complete perversion of the term. On the other 
hand, some men of the schools spend their energies in metaphysical nothings, 
splitting hairs in logic, and calling it scientific ; this is equally a perversion. 




222 NAVIGATION. 

positions, the moon is depressed by parallax, and appears nearer to 
those stars that are below her, and further from those stars that are 
above her, than would appear from the center of the earth. There- 
fore, the apparent altitudes of the two objects, must be taken at the 
same time that their distance asunder is measured. The altitudes 
must be corrected for parallax and refraction, thus obtaining the 
true altitudes. 

The annexed figure is a general representation 
of the triangles pertaining to a lunar observation. 

Let Z be the zenith of an observer, S' the 
apparent place of the sun or star, and aS its true 
place. Also, let m' be the apparent place of the 
moon, and m its true place as seen from the cen- 
ter of the earth. 

Here are two distinct triangles, ZS'm', and ZSm. The apparent 
altitudes subtracted from 90°, give ZS' and Zmf, and S'm' is the 
apparent distance ; with these three sides, the angle Z can be found. 
Correcting the altitudes, and subtracting them from 90°, will give 
the sides ZS and Zm ; these two sides, and their included angle at 
Z, will give the side Sm, which is the true distance. 

The definite true distance must have a definite Greenwich time, 
which can be readily found ; and this, compared with the local time 
deduced from an altitude of the sun, will of course give the longitude. 
We shall now make a formula to clear the distance. 

Let >S'=the apparent altitude of the sun or star, 
and #=the true altitude. Also, 
Let m'=the apparent altitude of the moon, 
and m =the true altitude. 

Observe that the letters with the accent, indicate apparent, and 
without the accent, the true altitudes. 

Put d to represent the apparent distance, and x to represent the 
true distance. 

Bear in mind, that the sine of an altitude is the same as the cosine 
of its zenith distance, and conversely, the sine of a zenith distance 
is the same thing as the cosine of the corresponding altitude. 

Now, by the fundamental equation of spherical trigonometry noted 
in the last chapter, we have 



LUNAR OBSERVATIONS. 223 

cos. d— sin. S' sin.m' A1 A „ cos. x — sin. S sin. m 

. Also cos. Z^ . . 

cos. S' cos. m' cos. & cos. m 

cos. d — sin. S' sin. m' cos. x — sin.S sin. m 



Whence, 

cos. S' cos. m' cos. <S cos. m 

By adding unity to each member we have 

i j_ cos - d — sin. S' sin. m' , . cos, x — sin. S sin. m 

cos. S' cos. m' cos. & cos. to 

(cos. S' cos. m' — sin. S' sin. w')-}-cos. rf (cos. S cos. m — sin. S sin. m)-f-cos. x. 
cos. S' cos. m' cos. iS cos. m 

By observing equation 9, plane trigonometry, we perceive that the preceding 
equation reduces to 

cos. (<S'-j-m')-}-cos. d __cos. (S-f-m)-j- cos. x 
cos. <S" cos. m! cos. S cos. m 

Whence cos. #=(cos.(S'-fm')+-cos.d) cos - gfcosm _ cog . (g-l-m). 

COS. S COS. w' 

It is here important to notice that the moon's horizontal parallax 
given in the Nautical Almanac, is the equatorial horizontal parallax; 
that is, it corresponds to the greatest radius of the earth. The 
diameter of the earth through any other latitude is less, and of 
course the corresponding parallax is less. 

We therefore give the following table for the reduction of the 
equatorial horizontal parallax, to the horizontal parallax of any 
other latitude ; it is computed on the supposition that the equatorial 
diameter is to its polar as 230 to 229. For example if the horizon- 
tal parallax in the Nautical Almanac is 55' — , in the latitude of 40° 
the reduction would be 6", and the parallax reduced would be 
54' 54", and if the parallax from the Nautical Almanac were 60' 
the reduction would be6 vv 6,and reduced would be 59 v 53". 4. 

The semi- diameter of the moon given in the Nautical Almanac is 
her horizontal semi-diameter, but when she is in the zenith she is 
nearer to us by the whole radius of the earth, about one-sixtieth 
part of her whole distance, consequently she must appear under a 
larger and larger angle as she rises from the horizon, and this is 
called the augmentation of the semi-diameter. 

We give the reduction for the parallax ; and the augmentation for 
the semi-diameter in the following tables : 



224 



NAVIGATION. 



Red. of 

Lat-" " 
itude. 


1 's Eq. hor. parallax. 


Eq. par. 
55' 


Eq. par. 

60' \ 


20° 


0".9 


1" 


25 


.2 .8 


3 


30 


3 .7 


4 


35 


4 .6 


5 


40 


6 .0 


6 .6 


45 


7 .3 


8 


50 


8 .6 


9 .4 


55 


10 .1 


11 


60 


11 


12 


65 


11 .8 


13 


70 


12 .8 


14 


15 


13 .9 


15 


80 


14 .6 


16 



Augmentation of the 


Moon's semfaliam. 


Ap. Alt. | Aug. 


6° 


2" 


12 


3 


18 


5 


24 


6 


30 


8 


36 


9 


42 


11 


48 


12 


54 


13 


60 


14 


66 


15 


72 


16 


90 


16 



We now give an example showing all the details of finding the 
longitude by a lunar observation. 

EXAMPLE. 

Suppose that on the 25th of January, 1 852, between three and four 
o'clock in the afternoon, local time, the observed distance between 
the nearest limbs of the sun and moon was 50° 3' 20", the altitude 
of the sun's lower limb was 20° 1', and of the moon's lower limb 
48° 57', height of the eye 16 feet. The latitude corrected for the 
run from noon was 34° 12' N. 9 and the supposed longitude about 
65° west. What was the longitude ? (the Nautical Almanac being 

at hand.) 

Preparation. 



Supposed time at ship, 
Supposed longitude 65 
Supposed time at Greenwich, 



n. M. 

3 15 P. M. 

4 20 



7 25 P. M. 

On the 25th at noon the N. A. gives the ®'s S. D. at 14' 47", and at midnight 
at 14' 45" 7 ; therefore at the time of observation we take it at 14' 46'', by simple 
inspection. In the same summary manner we take the Equatorial horizontal 
parallax at 54' 12". 



©'s semi-diameter, - 
Aug for Alt. - 

f)'s true S. D. - 
Observed distance, - 
Sun's S. D. 
Moon's S. D. 
Apparent central dis. 



- 


14' 46" 


- 


12 


- 


14' 58" 


50° 3' 20" 


16 


16 


14 


58 


50° 34' 


34"=d 



$)'s Eq. hor. par. 
Red. for lat. - 

Reduced hor. par. 

Alt. O's L L 48° 57' 
O's S. D. 14' 58" 

Dip — 3 56 



54' 12" 
4 

54" 8" 



©'s app. alt. 49° 8' 2" 



LUNAR OBSERVATIONS. 



225 



Alt. 0' s lower L 

Semi-diameter, 

Dip, 

O's app. alt. 

Refraction, 

O's true alt. 



20° r 
-1-16' 16" 

— 3' 56" 
20° 13' 12"=S' 

— 2' 34" 
S. 



20° 10' 38' 
(J)'s app. alt. 
Parallax in alt. 
Refraction, 

True alt. 
(S'-\-m')=69° 21' 14" 



N, B. To find the moon's parallax in 
altitude see problems on page 201. 



f)'s app. Alt. 49° 8' 
54' 8"=3248' 
36' 2"=2120 



49° 8' 2" 
35' 20 
—49 



ocs. 9.815778 
log. 3.511616 

3.326394 



49° 42' 35"=wi 
(AS y +m)=69° 53' 53". 
We are now prepared to apply the equation to compute the true 
distance. The equation requires the use of natural sines and 
cosines. 

cos. z=(cos. (5f'+!»')+cos. d) C0S 'f cos - m — cos. (S+m) 

cos. aS cos. m' 

(£'+wi') = 69°21'14"N.cos. .35259 
d=5Q° 34' 34" N. cos.* .63449 

.98708 log.— 1.994350 
£=20° 10' 38" log. cos. 9.972496 
m=49° 43' 15" log. cos. 9.810578 
£'=20° 13' 12" cos. com. 0.017626 
m'=49° 8' 2" cos. com. 0.184228 
Num. .97561 log. —1.989278 

=sum less 20. f 
N.cos. (S-\-m)=69° 53' 53" —.34369 
True distance, 50° 48' 29" cos.63 1 92 

In the Nautical Almanac, we find that at 6 P. M. mean Green- 
wich time, on said day, the true distance between the sun and moon 
was 49° 59' 26", and at 9 P. M., the distance was 51° 20' 49", show- 
But the change 



ing a change of 1° 21' 23" in three hours of time. 



* When d is greater than 90° its cosine becomes minus, and its numerical 
value is then the natural sine of the excess over 90°. Thus if d were 105°, its 
cosine would be numerically equal to the sine of 15°, and must then be subtracted 
from the cosine of the sum of apparent altitudes. The result (cos. x) would 
then be the sine of the excess over 90°. 

t Less 20 because the table of natural sines is to radius unity, and we used 
cos. S and cos. m to the radius of 10, making two tens to take away. 
15 



226 NAVIGATION. 

from 49° 59' 26" to 50° 48' 29" is 49' 3" ; and now on the suppo- 
sition that the change is in proportion to the time ( and it is very 
nearly ), we have the following analogy 

1° 21' 23" : 49' 3" : : 3k. : t 
Or, 4883 : 2943 : : 3 : lh. 48m. 295. 

That is, the time that this observation was taken lh 48m 29s after 
6 at Greenwich. Or, 7h 48m 29s mean Greenwich time. 

With the true altitude of the sun 20° 10' 38", the latitude 34° 
12', and the polar distance 109° 0' 48", we find the apparent time 
at ship 3h 10m 5s, to which we would add the equation of time, 
12m 34, making the mean time 3h 22m 39s. 

From the Greenwich time lh. 48m. 295. 

Sub. time at ship - - 3 22 39 

Giving Ion. in time - 4 25 50=66° 27' 38" W. 

West, because the time at Greenwich was later in the day. 

If a lunar is taken with a star, or with the sun, when the sun is not 
in a proper position to depend upon its altitude for local time, the 
time must be noted by a watch, and the difference between the watch 
and true time made known, by a previous or subsequent observation 
on the sun, or some star which is nearly east or west of the observer. 

The most material part of working a lunar is that of clearing the 
distance. We, therefore, give the following examples, without the 
little incidental details. 

We show the working of one in which the distance is greater 
than 90°. 

The apparent distance between the center of the sun and moon 
on a certain occasion, was 98° 12' ; the apparent altitude of the 
sun's center was 54° 10', and of the moon's 20° 37' ; the moon's 
horizontal parallax at the same time was 57' 12". What was the 
true distance ? Ans. 97° 25' 10" 

Horizontal par. 57' 12"=3432 log. 3.532547 
D Alt. 20° 37' cos. 9.971256 

Parallax in alt. 53° 31'=3211 log. 3.586803 



LUNAR OBSERVATIONS 



227 



Q)'sapp. alt. 20° 37' 
Refraction —2' 31" 

Parallax +53' 31" 



^ app. alt. 54° 10' 0" 
Refraction —40" 



's true alt. 54° 9' 20" 



Q)'s true alt. 21° 28' 

(£'_j_ m ') = 74° 47' 

Nat. cosine 74° 47' is -(-.26247 

tf=98° 12' Nat. cos. is —14263 



(£-|-m)==75 37' 20" 



Algebraic sum 

#=54° 9' 20" cos. 

m=21° 28' cos. - 
>S"=54° 10' cos. complement 
wi'=20° 37' cos. complement 



.11984 log. 



•1.078602 
9.767591 
9.968777 
0.232525 
0.028744 



Sum (rejecting 20) —1.076239 
This log. corresponds to -f-. 11919 

—Nat. cos. (>S4-m)=cos. 75° 37' 20" —.24832 

cos.*=cos. 97° 25' 10" —.12913 

N. B. The last Nat. cosine having the minus sign shows that 
the corresponding arc must be greater than 90°. To find the 
arc we conceived .12913 to be plus, and found it corresponded 
in the table to the Natural sine of 9° 25' 10", and to this we 
added 90° for the result. 

EXAMPLES FOR PRACTICE. 



No. 
1 


Ap. alt. of sun 
or a fixed star. 


Moon's ap. 
altitude. 


Apparent cen- 
tral distance. 


Moon's 
hor. par. 


True 
distances. 


8°6 3 


39 18 


O 1 II 

46 45 


53 51 


O 1 II 

46 4 25 


2 


# 29 47 


57 22 


27 35 


60 3 


28 8 24 


3 


O 31 14 


28 7 


14 21 30 


54 29 


14 9 24 


4 


O 60 5 


63 12 


51 321 


58 30 


50 41 15 


5 


* 34 28 


10 42 


49 18 38 


61 11 


48 45 39 


6 


O 8 26 


19 24 


120 18 46 


57 14 


120 1 46 


7 


* 43 27 


40 9 


18 21 35 


60 20 


18 8 12 


8 


* 53 13 


57 32 


60 13 49 


60 52 


59 48 12 


9 


O 72 26 


18 30 


81 2 28 


60 58 


80 9 33 


10 


O 60 33 


9 26 


70 36 16 


59 57 


69 49 12 



228 navigation:. 

The foregoing method of clearing a lunar distance is very 
good, as an educational exercise, but for practical use, it is ob- 
jectionable, as the equation requires the use of natural sines and 
cosines. To ensure a complete understanding of this important 
subject, theoretically and practically, we will further transform 
the equation 

/ ct7~i 7 i ?\ GOS.S COS.Wi /ci\ \ t a \ 

cos. x= ( cos. JS -\-m-\-cos.a) _ — cos.f S4-m), ( 1 ) 

cos.S' cos.m' 

and adapt it to the use of logarithmic sines and cosines. 

Conceiving (S'-\-m') to be a single arc, and applying equation 
(17) (page 50), to the first factor in the second member of (1), 
we shall have 
cos.#= (£) 

2coa ± (S f - \ -m'- \-d)co8±(S'-\-m' — d)cos.S cos.m , ~. . 

cos.S' cos.m' 
By equation (32) (p. 51), we find that cos.#=l — 2sin. 2 \x. 
By eq. (31) (p. 51), cos.(^4-m)=2cos. 2 i(^+w)— 1. 

These values of cos.a; and cos.^-j-m), placed in (2) will give 

I o . 2 ± 2cos.^(^ / -}-w / -[-^)cos.|(AS f/ -{-m / — d)cos.Scos.m 

cos. $' cos.m' 
+1— 2cos. 2 -»-(/S4-m). 
By dropping the units in each member, and dividing by — 2, 
we have 
sin. 2 ^= (3) 

COS./S' COS.m' 

By division, we obtain 
sin. 3 \x 



cos. 2 ftS+m) 

1— cos±(S'+m'+d)cos.i(S'+m'— d)eos. S ^ 5m (4) 

cos. 2 |(#-]-m)cos.$' cos.m' 
Assume 

. 8 p_ cos.| (S'-\-m'-\-d)cos.±(S'~\-m f — <#)cos. S cos,m ,-x 
cos. 2 |( S-]-m)cos.S' cos.m' 
Calling P an auxiliary arc, equation (4) now becomes 

s i^!_^_ = l--sin. 2 P. 

cos. 2 i(£+m) 



LUNAR OBSERVATION'S. 229 

Because sin. 2 P-J-cos. 2 P=l, cos. 2 P=l — sin. 2 P. 

iiri, sin. 2 i# „ „ 

Whence ? =cos. 2 P. 

cos. 2 i( S-\-m) 

By extracting square root and clearing of fractions, we have 

sin.£#=cos.Peos.£(/S'-f-wi). (6) 

Equations (5) and (6) are plain and practical, they can be 
easily remembered, and they are adapted to logarithms. 

Equations (5) and (6) can be put in words, and called a rule, 
but in our opinion this is not necessary. 

We will now re-compute the last example, in which 
S '=54° 10', £=54° 9' 20", m'=20° 37', m=21° 28', 

c?=98°22', (£'+m')=74°47'. 

i(£"+m'+6?)=86° 29' 30" cos. (less 10) —2.786704 

i(S'+rn'— tf)=ll°37'30" cos. " —1.991000 

S=54° 9' 20" cos. " —1.767598 

ra=21° 28' cos. —1.968777 

i(£+wi)=37° 48' 40" cos. complement 0.102364 

*cos. complement 0.102364 

£'=54° 10' cos. complement 0.232525 

m'=20° 37' cos. complement 0.028744 







2)— 2.980076 


Add 


=48 


—1.490038 
10 


sin.P 18° 0' 7" 
cos.P 9.978203 
cos ±(S+m) 9.897636 


9.490038 


sin.ia;= 9.875838= 


3 42' 25" 

2 



True distance, 97° 24' 50" 

The two methods do not give the same result precisely. But 
this one is the most reliable of the two. 

* The preceding log. repeated to obtain the square of the last quantity. 



APPENDIX. 



It is comparatively an easy matter, to conduct a survey, or navigate 
a vessel, when there are no important difficulties to be overcome ; but 
the true test of knowledge or skill in any pursuit, is to be found only in 
real adversity. The mariner who successfully manages his ship, when 
every thing is provided, when all is in order, and the weather favorable, 
is deserving of little credit ; but let the ship become disabled, and the 
storm terrific, and then there is scope for the exercise of every neces- 
sary acquirement, and its kindred talent. 

So it is with the man of science ; when every instrument is at hand, 
and all in order, it requires little skill, and but common knowledge, to 
make observations and experiments ; but when we reverse the case; the 
tact, knowledge, and ingenuity of the man, may oft times more or less 
overcome the difficulties. 

For instance, suppose it were necessary to find the altitude of the sun, 
for the purpose of finding the latitude of the place (on shore), or for the 
purpose of finding the time ; and we had no sextant or quadrant, and in 
fact, no instrument to measure angles. It could be done approximately 
as follows : 

Let a plumb line be suspended in water ; have a knot in the line, and 
let the knot be at a known distance above the water. The knot will 
cast a shadow on the water ; measure the distance of this shadow from 
the plumb line. The knot and its shadow, with the plumb line and 
water, will form a right angled triangle, and the angle at the base, com- 
puted by plane trigonometry, will be the altitude of the sun's upper limb, 
and this altitude may be used for any purpose, the same as if it were 
measured by a sextant, but the accuracy is not to be depended upon, for 
the want of delicacy in the instrument. 

A person on shore having a good watch, and knowing his latitude, 
can regulate his watch, or at least determine its rate and error for a 
short period of time. Then, if he have a nautical almanac, the common 
tables of logarithms, and a knowledge of spherical trigonometry, and a 



APPENDIX 



231 



corresponding knowledge of astronomy, he can find the longitude by a 
lunar observation, without a sextant, as follows: 

By the means of his watch and a plumb line, he will be able to range 
off an approximate meridian line. He will then observe the transits of 
stars, and of the moon across that meridian, taking those stars which 
are at that time near the moon's meridian, some to the east and some to 
the west of the moon, and some more north, and others more south 
than the moon. 

He will note the difference in time, between the transit of each star 
and the moon, across his approximate meridian, and by a combination, 
or rather comparison of these observations, he will be able to determine 
the moon's right ascension very nearly. By the moon's right ascen- 
sion, and the aid of the nautical almanac, he can find the Greenwich 
time. 

The Greenwich time, compared with the local time, will give the 
longitude. 

When we can find the moon in a vertical plane with any two fixed 
stars, and it be at the time the moon changes her declination very 
slowly, so that we can depend upon a declination taken from the nauti- 
cal almanac for the supposed time, we can then determine the moon's 
right ascension, and from thence the longitude as before, whether we 
are on land or sea. 

Ship-wrecked mariners, and travelers similarly situated, have fre- 
quently resorted to these artifices to obtain their approximate localities. 

We have frequently remarked in the course of this work, that the 
best position of a celestial object, at the time of taking its altitude, for 
the purpose of more exactly defining the time, is when the object is 
nearly east or west ; we now propose to show this conclusively, and 
therefore give the following 

INVESTIGATION. 

To find under what circumstances, in a given latitude, a small mistake in 
observing or correcting the altitude of a celestial object, will produce the 
smallest error in the time computed from it. 
Let Z be the zenith, P the pole, r the supposed place, and m the true 

place of the object. Let ms be a parallel of altitude, join the points m 

and r, and let pq be the arc of the equator 

contained between the meridians Pm and 

Pr. 

Then as Pm and Pr are equal, mr may 
be considered as a small portion of a par- 
allel of declination rs will be the error in 




232 APPENDIX. 

altitude, and pq the measure of the required error in time. And as the 
sides of the triangle msr will necessarily be small, that triangle may be 
considered as a rectilinear one, right angled at s ; and because the angle 
Prm is also a right angle, the angles smr and PrZ, being each the com- 
plement of mrs, are equal to each other 

We now have, 

rs : mr : : sin. smr (ZrP) : rad. (1) 

Also, mr : pq : : cos. qr : rad. (2) 

Multiplying these two proportions together, omiting the common 
factor mr, gives, 

rs :pq : : cos. qr sin. (ZrP) : (rad.) 2 (3) 

But, sin. rP or cos. qr : sin. rZP : sin. ZP : sin. (ZrP) (4) 
Whence, cos. qr sin. ZrP=sin. rZP sin. ZP (5) 

The first member of equation (5), is the same as the third term in 
proportion (3) ; therefore, proportion (3) may be changed to the 
following, 

rs :pqi : sin. rZP sin. ZP : (rad.) 8 

Whence, pq= ( !*£!*£ ) —l— 

r * \ sin. ZP I sin. rZP 

Now, as the quantities in parentheses are supposed to be constant the 

value of pq, the error in time must vary as varies ; and it is 

sin. rZP 

obvious that pq will be least, when sin. rZP is greatest, that is, when 

rZP=90°, or the object due east or west. 

Again, we can come to a like result, more directly and ele- 
gantly, by the direct application of the differential calculus. 

Let PZr be the spherical triangle, from which time, or the 
angle ZPr, is computed. This angle will vary as the altitude 
varies, ZP and Pr, the co-latitude and polar distance, being con- 
stant for any small portion of time. 

Let A represent the sun's true altitude, L the latitude of the 
observer, D the sun's polar distance, and P the angle at the 
pole, included between the meridian of the observer and the 
meridian of the sun. 

Now, by a fundamental equation in spherical trigonometry, 

we have 

r> sin. A — sin. L cos. D /C( n1 . % 

cos.P= -—. — _ (Seepage 214.) 

vcos.i/sin.i) v * & y 

Now the altitude of the sun, A, varies every instant, and this 



APPENDIX. 233 

causes the value of P to vary, but L and D are constants, 
therefore the differential of the above equation will be 

• -nj-n cos. AdA , , v 

—sm.PdP= ■. ■ ( 1 ) 

cos.Lsm.D 

But in the triangle ZPr we have 

cos. A : sin.P : : sin.Z) : s'm.Z. 

. n cos. A sin.Z /0 , 

sm.P= ^— (2) 

sm.U 

This value of sin.P placed in (1) will give 

cos. A sin.Z, n cos. Ad A 
dP=. 



sin.D cos.L sin. 2) 

Reducing, we find ~dP= d ~. 

cos. L sin.Z 

Now as cos.Z is a constant quantity, the value of — dP 9 or 
the second member, will be least when sin.Z is greatest. That is, 
when Z is a right angle, and the sun due east or west of the 
observer. 

The minus sign before dP shows that when A increases, P 
decreases, which is obviously true at all times. 

When Z=0, that is, the observer on the equator, the result 

dA 
will be — dP=— , and, if we suppose the sun, also, on the 

equator, it will all the while be either east or west of the ob- 
server, and then — dP—dA ; that is, the time and altitude would 
then have equal variation. 



LUNAR OBSERVATIONS, 



The differential calculus will apply most beautifully to the 
clearing of lunar distances from the effects of parallax and re- 
fraction. 

In this case we must regard the difference between the true 
and apparent altitude of the moon, as a differential quantity, 
and the refraction of the sun or star a differential quantity, and 
the difference between the true and apparent distance is a cor- 
rection sought, and it is also to be regarded as a differential 
quantity. 



234 



APPENDIX. 



Let ZS'm' represent the observed triangle. 
The observed center of the sun or star is at 
S', but the center really is at S. The ob- 
served center of the moon is at m', the real 
center i at m, as seen from the center of the 
earth. S 'm! is the apparent distance, but the 
true distance is S?n. 

Let S= the apparent altitude o the sun or star. 
m= the apparent altitude of the moon. 

And#= the apparent or observed distance S'm'. 

Now by spherical trigonometry, (see page 223 of this book,) 
we have 

cos. a; — sm.S sin.m 




cos.Z=- 



(i) 



cos.S cos.m 

In this problem the angle Z is always a constant quantity, S 
and m are variable, and x varies in consequence of the variations 
of S and m. But we may take these effects separately. That 
is, by supposing m only to vary, anol discover the corresponding 
variation for x. Then we may suppose S to vary, and obtain the 
corresponding variation to x ; and lastly, these two effects put 
together will be the total variation for x, or the difference between 
the apparent and true distance between the sun and the moon 
or a star and the moon, as the case may be. 

We will therefore differentiate (1) on the supposition that x 
and m are variables. 

That is, d. cos.Z cos. S cos.?n=d. cos.# — d. sin. £ sin.m. 

Or — cos.Z cos. S sin. mdm= — sin.x dx — sin. S cos.m dm. 

dx 
sin.z — =cos.Z cos./S' sin.m — sin. S cos.m. (2) 
dm 



But equation (1) gives cos.Z cos. #= 
Multiply by sin.m, then 



cos.rr — sin. S sin. m 



cos.m 



cos.Z cos. S sin. m= 



cos.z sin.m — sin.£sin. 2 m 
cos.m 



This value placed in equation (2), that equation becomes 



dx 



cos.r sin.m- 



-sin.#sin. 2 m 



dm 



— sin. # cos.m. 



cos.m 



APPENDIX. 235 

Or co_.77isin.# — =cos.# sm.m — sin.S sin. 2 m — sin./S'cos. 2 wi. 
dm 

=cos.#sin.m — sin.AS'(sin. 2 m-|-C05. 2 w). 
But (sin. 2 m-(-cos. 2 m)=l. Therefore 

dx . ct 

cos.msm.a; — =cos.# sin.m — sm.o. 
dm 

Or ^^/ cos^sin.m- sin^N^ fa 

\ cos.msm.x J 

Now if we suppose that x and S are the variables, in place of 
x and 77i, the result will be the same as (3) if we change S to m 
and m to S. 

Therefore /cos.* sin.ff-sin.mX^ .. ^ yalue of ^ copres _ 
\ cos. S sin. a; / 
ponding to the variation of the sun or star's altitude. 

The apparent place of the moon is below its true place, and 
the apparent place of the sun or star is above its true place, 
therefore dm and dS must have contrary signs. Consequently, 
the whole variation of x, when both S and m vary, (as is always 
the case,) must be 

, /cos. x sin. m — sin.^X, /cos. x sin. S — sin.m\,~ ,.* 

\ cos.msin.# / \ cos. £ sin. # / 

When the sun or star is at the zenith, (dS) is then nothing, 
and the value of dx is expressed by the first term of the second 
member. When the moon is in the zenith, then (dm) becomes 
nothing; but in practice, such cases would not be likely to occur, 
once in a life time. 

We will work the fourth example by this formula : 

Given, the sun's apparent altitude 60° 5'. The moon's apparent 
altitude 63° 12'. The apparent distance 51° 3' 21", and the moon's 
horizontal parallax 58' 30", to find the true distance from center to 
center, as seen from the center of the earth.* 

Ans. 50° 41' 15". 

•Correction may be made for the figure of the earth by correcting the 
parallax for the latitude of the observer, as shown in table on page 224. 



36 APPENDIX. 

Here £=60° 5', m=63° 12', 
Q) h. p.=58' 30"=3510", log. - 

cos. m 


s==51° 3' 21". 
3.545307 
9.654059 


Parallax in altitude +1582" 
Refraction —29" 


3.199366 


c?ra=+1553" dS= — 32" sun's refraction 


For the coefficient of dm, 

sin.m — 1.950650 cos.m 

cos.# — 1.798351 sin.rc 


—1.654059 
—1.890843 


.56108 —1.749001 
N. sin./S— .86675 


—1.544902 Den. 


—.30567 log. - 
dm 1553 log. - 


—1.485265 Num. 
—1.940363 
3.191171 


First part of dx — 1354", 


3.131534 


For the coefficient of dS, 

sin./S —1.937895 cos./S 
cos.s — 1.798351 sin.z 


—1.697874 
—1.890843 


+.54480 —1.736246 
N. sin.m— .89259 


—1.588717 Den. 


—.34779 log. - 


—1.540078 Num. 


dS=— 32" log. 

28"6 
Whence dx=— 1354"+28"6= —22' 6" 
Apparent distance, 51° 3' 21" 


—1.951361 
+1.505150 

1.456511 


True distance, 50° 41' 15" 




The equation 

/7/i# _(cos.a; sin.ra — sin./S)^ w /cos.x sin 


./S-sin.m\^ 



can be put into another form, which will better suit the tastes 
of mere practical men, and avoid the use of natural sines. 



APPENDIX. 237 

Assume cos.z sin.wi=sin.^4, and cos.#sin.£=sin.2?, and 
determine the values of A and B. Then we shall have 

cos.msin.x \ cos.£sin.# / 

The first term of the second member will be plus or minus, 
according as A is greater or less than S. The coefficient of dS 
is positive, when B is greater than m. But (dS) is always nega- 
tive ; hence, the product will be positive or negative, according 
to the rules of algebraic multiplication. 

By the application of equation (16), page 50, and dividing by 
2, the preceding equation becomes 
%dz = 

cos.±(A-\-S)s'm.±(A^S) , cos.^(B-\-m)sm.\(B ^m) jg 
cos.msin..r cos.£sin.a; 

This equation, put in words, would be a rule for clearing the dis- 
tance, bat those who comprehend it, can follow it as readily with- 
out the words, as with them. We illustrate by a single example. 
The sign <S> indicates the difference between the two quantities between 
which it is placed, when it is not known which is the greatest. 

EXAMPLE. 

Suppose that in latitude 46° north, the moon's apparent altitude 
teas 36° 28', and that of a planet 24° 43', and their apparent dis- 
tance asunder was 71° 46' 24". The moon's horizontal parallax at 
that time was 58' 31", and that of the planet 29"; what was the true 
distance as seen from the center of the earth? 

PREPARATION. 

Q)'s hor. par. 58' 31" 
Table, page 224, 7"7 



58' 23"3=3503.3 log. 3.544477 
cos.Q)'s alt. 36° 28' 9.905366 



Q)'s parallax in alt. 2817" 3.449843 

Refraction - — 77 

c?w=2740" 



238 



APPENDIX. 



Planet. 
Log. 29" 1.462398 
Cos. 24° 43' 9.958271 



Parallax in Alt. 
Refraction, 



+26*3 
-124 



1.420669 



dS — 97"7 

Here w=36° 28', 5=24° 43', and *=71° 46'.* 

For the auxiliary arcs A and B, 

cos.z 71° 46' 9.495388 - - - 9.495388 
sm.m 36° 28' 9.774046 sin.tf 24° 43' 9.621313 



A 
S 



10° 43' 9.269434 
24° 43' 



7° 31' 9.116701 
36° 28' 



J 2 (S+A) = 17° 43' 
±(S—A)= 7° 0' 

We now follow the formula. 

1st Term. 

cob.±( S+A) 17° 43' 9.978898 

sin.i(#— A) 7° 0' 9.085894 

cos. compl.m 36° 28' 0.094634 

sin. comply 71° 24' 0.023298 

dm 2740" 3.437751 



±(m+B)=2l° 59' 
i(m— £)=14°28' 



2nd Term. 
cos.\(m+B) 21° 59' 9.967217 
sin4(m— B) 14° 28' 9.397621 
cos. comple. 24° 43' 0.041729 
sin. comple. 71° 24' 0.023298 
dS — 97"7 1.989895 



(Less 20). — 417"3 2.620475 



+26"29 1.419760 



The first term is minus, because A is less than S, and the 
second term, is plus, because it contains the product of two 
minus factors, (sin.i? — sin.wi) and dS. 

Whence ±dx=— 417.3-f26"3=— 391" 

Or <&=— 782"= - - —13' 2" 

Apparent distance, - - - 71° 46' 24" 

True distance, - - - - 71° 33' 22" 



* In computing the coefficients, seconds of arc need not be noticed. 



APPENDIX. 



239 



We give the following examples of distances between the 
moon and planets, for exercises : 





Moon' 


s Appa- 


Planet's Ap- 


Moon's Dist. 


Moon' 


sHor. 


Planet's 


True 


No. 


rent Altitude. 


parent Alt. 


from Planet. 


Parallax. 


Parallax. 


Distance. 




o 


/ 


o / 


O t ll 


/ 


ii 


ii 


O i ll 


1 


58 


36 


16 23 


69 37 20 


56 





31 


69 40 30 


2 


80 


4 


35 30 


60 4 3 


61 


16 


18 


59 58 57 


3 


16 


26 


29 41 


98 15 31 


60 


35 


30 


97 45 4 


4 


50 


14 


51 3 


40 


54 


50 


25 


39 44 42 


5 


62 


12 ( 


38 27 


37 50 34 


55 


13 


23 


37 58 14 



LO GARITHM S. 

In the forepart of this volume, we have shown the practical 
uses of logarithms, using the common tables, which extend only 
to six places of decimals, and this is sufficient for all common 
purposes. But for those who desire to be more nice and accu- 
rate, we have computed a table extending to twelve decimal 
places, including the consecutive numbers from 1 to 200, and 
from thence, the prime numbers as far as 1543, together with 
the Auxiliary Logarithms of unity, and a fraction as low as the 
number 1.0000000001. 

In these logarithms the index is omitted, as it is not necessary 
when one has obtained the true theory of logarithms. For in- 
stance, the log. of the number 28 has a certain decimal part, 
which must remain the same if that number be changed to 2.8, 
or to .28, or to 280, &c. &c, and according to the value of the 
2 and 8, the operator will prefix the index. 

To make a table of logarithms anew, to contain any particular 
number of decimal places, the following formula, taken from 
algebra, is the most practical and convenient of any yet known. 

Log.(*+l)= 

log.2+.8685889638^ — — + \ + \ \ka. 

5 T V 22+1^3(22+1)^5(22+1)*/ 

This formula will give the log. of (2+1) when the log. of z 
is known, but the log. of 2 is known when we make 2=1, 10, 
100, 1000, <fec. Then the formula will give the logarithms of 
2, 11, 101, 1001, &c. 



240 APPENDIX. 

When z is large, over 100, the series converges very rapidly, 
and then, only two terms need be used. When z is over 
2000, only two terms need be used, even for twelve decimal 
places. 

The auxiliary logarithms, A, B, C, page 71 of tables, were 
computed by this formula. For instance, the log. of 1001 is 
the same in its decimal part as the log. of 1.001. Hence, when 
we have the logarithms of 1001, 1.002, 1003, &c, we have the 
logarithms of 1.001, of 1.002, &c. 

The greater the number the more readily can its logarithm be 
computed. 

That the learner may fully comprehend the application of 
these auxiliary logarithms, A, B> C, he must call to mind the 
following principle : 

(Art. 14.) The product of any number of factors consisting of 
unity and a small fraction, is very nearly equal to unity, und the 
sum of those fractions . 

Thus the product of (1.0001), (1.00002), (LOOOWM), is very 
nearly equal to 1.000123. 

If this be true, we can immediately separata i. 000123 into 
the same factors. 

The number 1.00021 may betaken fortheproAuctof (1.00004) 
(1.00017), without any material error. 

Or (1.00021) maybe taken for the product of 21 factors, each 
equal to (1.00001), with very little error; and if this be true, 
1.00001 is the 21st root of 1.00021 very nu*rly. 

This principal may be proved algebraically thus : Let a, b t 
and c, be very small fractions, then the product of ( l+«) • 0+^) 
is \-\-a-\-b-\-ab, as we find by actual multiplication. 

But a and b being very small fractions, their product ab is 
extremely small in respect to a or b, and therefore ab may be 
omitted, and the essential part of the product is \-\-a-\-b, that is, 
unity and the sum of the fractions. 

Now let {a-\-b) be s, and the product of (1+5) into (1+c) 
will be 1+s+c ; that is, the product of (1+a) (l+£) 0+ c ) 
cannot be far from 1 -f-a-|-5-f-c. 



APPENDIX. 241 

Let us try this in numbers. Multiply 1.0001 by 1.00004. 

1.00004 
1.0001 



100004 
1.00004 



1.000140004 

But this value is extremely near 1.00014, the sum of unity and 
the fractional parts of the factors. 

(Art. 15.) When the difference between two quantities of 
the same kind is very small in relation to the quantities them- 
selves, such a difference (in the higher mathematics) is some- 
times called a differential. 

Thus, in the last example, the difference between 1.000140004 
and 1.00014 is .000000004, and it is so small in relation to 
1.00014 that it may be omitted withont sensible error, and it is 
then a practical differential. 

The difference between 8 and 9 is 1, but in this case 1 cannot 
be a differential, it is too large. 

The difference between 80000 and 80001 is 1, and here 1 
might be taken as a differential in some practical computations, 
and therefore omitted. 

There is no exact line of demarkation where a difference may 
be taken for a differential ; that depends on the nature of the 
case ; hence, those who distrust their own judgments are gener- 
ally prejudiced against the calculus. 

If we take the logarithmic formula from (Art. 13,) and con- 
ceive z to be very large, then the difference between (z-|-l) and 
z is one (very small), and may be regarded as the differential of 
z: and in that case \og.(z-\-l) — log.2 is the same as the differ- 
ential of the logarithm of z. 

Making this supposition, the formula in (Art. 13) becomes 

(dif.) log.z=0.8685889638X-^^. 

(2*4-1) 

We take but one term of the series, because the following 
terms are of no essential value compared to the first, and as z is 
very large, (22-f-l) is comparatively so little greater than 2z, 
16 



242 APPENDIX. 

that for all practical purposes, it may be taken as 2z, and then the 
preceding equation will reduce to 

(^if ) W y _ 0-4342944819(dif.> 

Z 

The symbol (dif.) signifies the differential of the quantity 
which follows it, and this equation, put in words, gives the fol- 
lowing rule to adjust a logarithm to correspond to any particular 
number. 

Rule. — The differential of a logarithm is equal to the differential 
of the number multiplied by the modulus, and the product divided by 
the number. 

"When we wish to find the differential of a number corres- 
ponding to. a given differential of a logarithm, we change the 
equation to the following : 

(dit)z=-<^9}^±. (2) 

v ' 0.4342944819 V J 

This gives the following rule to correct a number : 
Rule. — The differential of a number is equal to the number mul- 
tiplied into the differential of the logarithm, and that product divided 
by the modulus of the system. 

The practical use of equations (1) and (2) will be found in 
the following 

EXAMPLES. 

1 . When the diameter of a circle is 1 , the circumference is 
3.14159265359. Find the logarithm of this number true to at 
least ten decimal places. 

The number being between 3 and 4, the index is 0. During 
the operation we shall pay no regard to the index of any loga- 
rithm we may take out, because it will not be necessary. We 
may consider the number to be 314 and a decimal, or we may 
take the whole for a whole number, but it is best to take 314, 

* This equation is found in the differential calculus, thus : dx= , an 

equation in which ar is a logarithm, y a number, and m the modulus. Here 
then, rough and practical as our equations appear to be, they exactly coin- 
cide with the most refined theory. 



APPENDIX 



243 



the three superior digits, as a whole number, whatever the number 
may be. 

Table III. commencing on page 67 of tables, and the auxiliary- 
logarithms following, are the log. referred to. 
314=157X2. 

Log. 157 - - - 0.195899652409 
Log. 2 - 0.301029995664 



Log. 314 
Table B, 1.0005 log. 



Product, 
Table C, 



Product, 
Given number, 



log. 



3141570 log. 
1.000007 

2 199099 
314157 



314159 199099 log. 
314159 265359=^ 



0.496929648073 
0.000217092970 

0.497146741043 
0.000003040047 



0.497149781090(a) 



Biff. 



66260=(dif.)z. 

We have log. (a) the logarithm of a number very near z, so 
near, that the difference may be called a differential ; therefore 
we may use equation (1). 

fdif.) W--( °-43429448)(66260 ) 

1 ' ° 314159199099 

But we may take common logarithms to reduce this second 
member. In that case we have only one log. to find, as the log. 
of the modulus is constant, and (a) may be used for log. of z. 



log. 66260 
log. m 

log. z - 

Num. 0.000000091596 
To (a) - 

Add dif. of z 



4.821251 
—1.637784 

4.459035 
11.497150 

—8.961885 
0.497149781090 
0.000000091596 



Num. 3.14159265359 log. —0.497149872686 Arts. 



244 



APPENDIX. 



The factor 1.0005 was obvious, but the other factor 1.000007 
may not be so, — the question then is, how did we obtain it? 
It was thus : 

3.14157#=3.14159265359( 
Whence x= 1 .000007+. 

2. The sidereal year consists of 365.2563744 mean solar days. 
What is the logarithm of this number? 

If we take the four superior figures, 3652, as a whole number, 
and factor it, we shall find that it is equal to 44*83. Hence 



Log. 44 - 
Log. 83 


- 0.643452676486 
0.919078092376 


Log. 3652 - 
(1.0001 log. (B) 
Factors i 1.00005 log. (G) 
(1.000004 log. (G) 


- 0.562530768862 

.000043427277 

21712704 

1737173 


365.2562408 log. - 
365.2563744 given number. 


- 2.562597646016 (< 



.0001336 differential. 
We obtained the above factors by solving the following equa- 
tion : 

3652^=365.2563744. 
Whence z=1.000154, (Art. 14,) (1.0001) (1.00005) (1.000004). 
To correct (a), and make it exactly correspond to the given 
number, we have the following expression : 

dif. loo- - (0.43429448)(1336) 
8 * 3652562408 . 



By common logarithms, 



Num. 0.000000158500, 
To (a) 
Diff. - ' - 

Log. sought, 



log. m 
log. 1336, 

log. a 



—1.637784 
3.125806 

2.763590 
9.562597 

—7.200993 
2.562597646016 
.000000158500 

2.562597804516 



APPENDIX, 



245 



3. When the radius of a circle is 1 , the natural sine of 7° 30' 
is expressed by the decimal 0. 1 30526 1921. What is the logarithm 
of this number? 



Log. .13 
Log. 1.004 

Product 



Factors 



.13052 
1.00004 log. 
1.000007 log. - 
1.0000004 log. 



Log. sin. of 7° 30' (Rad. unity,) 
For the common table, add 



0.113943352307 
0.001733712775 

0.115677065082 

- 000017371430 

3040047 

173716 

—1.115697650275 
10. 



4(») 



Log. sin. 7° 30', for table, - 9.115697650275 

By the foregoing, the reader will perceive that the logarithm 

of any number can be found by these tables, true to at least ten 

places of decimals. 

We are now prepared to take the converse problem, that is : 

Given a logarithm to find its corresponding number. 

EXAMPLES. 

1. What number corresponds to the log. 4.636747519487? 

Comparing the decimal, with the decimal logarithms in the ta- 
ble, we perceive that it corresponds to a number which is a little 
greater than 433 ; but as the index is 4, the real number must 
be a little greater than 43300. Let this be one factor of the 
required number, then 



From the given log. 
Subt. log. of 43300 



Log. 1 .0005, table B, 

Log. 1.00009, table C, 
Log. 1.000007, table C, 



4.636747519487 
4.636487896353 

0.000259623134 

217092970 next less 
in B. 



.000042530164 
39083266 



3446898 
3040047 



406851 



246 APPENDIX. 

.000000406851 
Log. 1.0000009, - 390861 9(») 

15990 
Log. 1.00000003 - - - 13029 3(o) 

2961 
The product of these small factors is 1.00059793. (Art. 14.) 
Multiply this product by the factor 43300. 

Or which is the same thing, 

Multiply 100.059793 

By 433 

300 179379 
3001 79379 
40023 9172 



Approximate number, 43325.890369 

But the last remainder in the logarithm (2961) may be taken 
as the differential of a logarithm, and corresponding thereto is 
a differential of the number, which must be added. 

It is found thus : 

Diff. of the number == C 433 ^)(.00 000000296 1 ) < 

.43429448 

By log. log. 43325 - - 4.636747 (the given log.) 
.000000002961 - - —9.471438 



—4.108185 
Log. .43429, &c. —1.637784 



Correction 0.0002954 log. —4.470401 

Approx. num. 43325.890369 

Num. sought, 43325.8906644 

EXAMPLES. 

1. What number corresponds to the log. 2.204923118054? 

Ans. 160.29616. 

2. What number corresponds to the log. 4.133409102? 

Ans. 13595.93. 

3. What number corresponds to the log. 3.2789020746? 

Ans. 1900.64967. 



LOGARITHMIC TABLES; 



ALSO A TABLE OF THE 



TRIGONOMETRICAL LINES; 

AND OTHER NECESSARY TABLES 





LOGAKITHMS OF 


1 

NUMBERS 




FROM 








1 TO 10000, 




N. 


Log. 


N. 


Log. 


N. 


Log. 


N. 


Log. 


1 


000000 


26 


1 414973 


51 


1 707570 


76 


1 880814 


2 


301030 


27 


1 431364 


52 


1 716003 


77 


1 886491 


3 


477121 


28 


1 447158 


53 


1 724276 


78 


1 892095 


! 4 


602060 


29 


1 462398 


64 


1 732394 


79 


1 897627 


; 5 


698970 


30 


1 477121 


55 


1 740363 


80 


1 903090 


6 


778151 


31 


1 491362 


56 


1 748188 


81 


1 908485 


7 


845098 


32 


1 505150 


67 


1 755875 


82 


1 913814 


! 8 


903090 


33 


1 518514 


68 


1 763428 


83 


1 919078 


9 


954243 


34 


1 531479 


59 


1 770852 


84 


1 924279 


10 


1 000000 


35 


1 544068 


60 


1 778151 


85 


1 929419 


11 


1 041393 


36 


1 556303 


61 


1 785330 


86 


1 934498 


12 


1 079181 


37 


1 568202 


62 


1 792392 


87 


1 939519 


13 


1 113943 


38 


1 579784 


63 


1 799341 


88 


1 944483 


14 


1 146128 


39 


1 591065 


64 


1 806180 


89 


1 949390 


15 


1 176091 


40 


1 602060 


65 


1 812913 


90 


1 954243 


16 


1 204120 


41 


1 612784 


66 


1 819544 


91 


1 959041 


17 


1 230449 


42 


1 623249 


67 


1 826075 


92 


1 963788 


18 


1 255273 


43 


1 633468 


68 


1 832509 


93 


1 968483 


19 


1 278754 


44 


1 643453 


69 


1 838849 


94 


1 973128 


20 


1 301030 


45 


1 653213 


70 


1 845098 


95 


1 977724 


21 


1 322219 


46 


1 662578 


71 


1 851258 


96 


1 982271 


22 


1 342423 


47 


1 672098 


72 


1 857333 


97 


1 986772 


23 


1 361728 


48 


1 681241 


73 


1 863323 


98 


1 991226 


24 


1 380211 


49 


1 690196 


74 


1 869232 


99 


1 995635 


25 


1 397940 


50 


1 698970 


75 


1 875061 


100 


2 000000 


N 


. B. In the following table, in the last ni 


ne columns of each p 


age, where 


the i 


irst or leading figures change from 9's 


to 0's, points or do 


;s are now 


intr< 


)duced instead of the 0's through the r 


jst of the line, to cat< 


;h ihe eye, 


and 


to indicate that from thence the corr< 


jsponding natural n 


umbers in 


thel 


first column stands in the next lower 


line, and its annexe* 


i first two 


figu 


res of the Logarithms in the second co 


lumn. 





LOGARITHMS OF NUMBERS. 3 




N. 





1 


2 1 3 


4 


5 


6 


7 


8 


9 




100 


000000 


0434 


0868 


1301 


1734 


2166 


2598 


3029 


3461 


3891 




101 


4321 


4750 


5181 


5609 


6038 


6466 


6894 


7321 


7748 


8174 




. 102 


8600 


9026 


9451 


9876 


.300 


.724 


1147 


1570 


1993 


2415 




103 


012837 


3259 


3680 


4100 


4521 


4940 


5360 


5779 


6197 


6616 




104 


7033 


7451 


7868 


8284 


8700 


9116 


9532 


9947 


.361 


.775 




105 


021189 


1603 


2016 


2428 


2841 


3252 


3664 


4075 


4486 


4896' 




10S 


5306 


5715 


6125 


6533 


6942 


7350 


7757 


8164 


8571 


89 V 8 




107 


9384 


9789 


.195 


.600 


1004 


1408 


1812 


2216 


2619 


3021 




108 


033424 


3826 


4227 


4628 


6029 


5430 


5830 


6230 


6629 


70-28 


i 


109 


7426 


7825 


8223 


8620 


9017 


9414 


9811 


.207 


.602 


.998 


\ 


110 


041393 


1787 


2182 


2576 


2969 


3362 


3755 


4148 


4540 


4932 


I 


111 


5323 


5714 


6105 


6495 


6885 


7275 


7664 


8053 


8442 


8830 


1 


112 


9218 


9606 


9993 


.380 


.766 


1153 


1538 


1924 


2309 


2694 


113 


053078 


3463 


3846 


4230 


4613 


4996 


5378 


5760 


6142 


6524 




114 


6905 


7286 


7666 


8046 


8426 


8805 


9185 


9563 


9942 


.320 


115 


060698 


1075 


1452 


1829 


2206 


2582 


2958 


3333 


3709 


4083 




116 


4458 


4832 


5206 


5580 


5953 


6326 


6699 


7071 


7443 


7815 




117 


8186 


8557 


8928 


9298 


9668 


..38 


.407 


.776 


1145 


1514 




118 


071882 


2250 


2617 


2985 


3352 


3718 


4085 


4451 


4816 


5182 




119 


5547 


5912 


6276 


6640 


7004 


7368 


7731 


8094 


8457 


8819 




4 
120 


9181 


9543 


9904 


.266 


.626 


.987 


1347 


1707 


2067 


2426 




121 


082785 


3144 


3503 


3861 


4219 


4576 


4934 


5291 


5647 


6004 




122 


6360 


6716 


7071 


7426 


7781 


8136 


8490 


8845 


9198 


9552 




123 


9905 


.258 


.611 


.963 


1315 


1667 


2018 


2370 


2721 


3071 




124 


093422 


3772 


4122 


4471 


4820 


5169 


6518 


5866 


6216 


6562 




125 


6910 


7257 


7604 


7951 


8298 


8644 


8990 


9335 


9681 


1026 




126 


100371 


0715 


1059 


1403 


1747 


2091 


2434 


2777 


3119 


3462 




127 


3804 


4146 


4487 


4828 


5169 


5510 


5861 


6191 


6531 


6871 




128 


7210 


7549 


7888 


8227 


8565 


8903 


9241 


9579 


9916 


.253 




129 


110590 


0926 


1263 


1599 


1934 


2270 


2605 


2940 


3275 


3609 




130 


3943 


4277 


4611 


4944 


5278 


6611 


5943 


6276 


6608 


6940 




131 


7271 


7603 


7934 


8265 


8595 


8926 


9256 


9586 


9915 


0245 




132 


120574 


0903 


1231 


1560 


1888 


2216 


2644 


2871 


3198 


3525 




133 


3852 


4178 


4504 


4830 


5156 


6481 


5806 


6131 


6456 


6781 




134 


7105 


7429 


7753 


8076 


8399 


8722 


9045 


9368 


9690 


..12 




135 


130334 


0655 


0977 


1298 


1619 


1939 


2260 


2580 


2900 


3219 




136 


3539 


3858 


4177 


4496 


4814 


6133 


5451 


5769 


6086 


6403 




137 


6721 


7037 


7354 


7671 


7987 


8303 


8618 


8934 


9249 


9564 


138 


9879 


.194 


.508 


.822 


1136 


1450 


1763 


2076 


2389 


2702 


139 


143015 


3327 


3630 


3951 


4263 


4574 


4885 


5196 


5507 


5818 




140 


6128 


6438 


6748 


7058 


7367 


7676 


7985 


8294 


8603 


8911 




141 


9219 


9527 


9835 


.142 


.449 


.756 


1063 


1370 


1676 


1982 




142 


152288 


2594 


2900 3205 


3510 


3815 


4120 


4424 


4728 


5032 




143 


5336 


5640 


5943 


6246 


6549 


6852 


7154 


7457 


7759 


8061 




144 


8362 


8664 


8965 


9266 


9567 


9868 


.168 


.469 


.769 


1068 




145 


161368 


1667 


1967 


2266 


2564 


2863 


3161 


3460 


3758 


4055 




146 


4353 


4650 


4947 


5244 


5541 


5838 


6134 


6430 


6726 


7022 




147 


7317 


7613 


7908 


8203 


8497 


8792 


9086 


9380 


9674 


9968 




148 


170262 


0555 


0848 


1141 


1434 


1726 


2019 


2311 


2603 


2895 




149 


3186 


3478 


3769 


4060 


4351 


4641 


4932 


5222 


5512 


6802 





4 


LOGARITHMS 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




150 


176091 


6381 


6670 


6959 


7248 


7536 


7825 


8113 


8401 


8689 




151 


8977 


9264 


9552 


9839 


.126 


.413 


.699 


.985 


1272 


1558 




152 


181844 


2129 


2415 


2700 


2985 


3270 


3555 


3839 


4123 


4407 




153 


4691 


4975 


5259 


5542 


5825 


6108 


6391 


6674 


6956 7239 




154 


7621 


7803 


8084 


8366 


8647 


8928 


9209 


9490 


9771 


..51 




155 


190332 


0612 


0892 


1171 


1451 


173Q 


2010 


2289 


2567 


2846 




156 


3125 


3403 


3681 


3959 


4237 


4514 


4792 


5069 


5346 


5623 




157 


6899 


6176 


6453 


6729 


7005 


7281 


7556 


7832 


8107 


8382 




158 


8657 


8932 


9206 


9481 


9755 


..29 


.303 


.677 


.850 


1124 




159 


201397 


1670 


1943 


2216 


2488 


2761 


3033 


3305 


3677 


3848 




160 


4120 


4391 


4663 


4934 


5204 


5475 


5746 


6016 


6286 


6556 




161 


6826 


7096 


7365 


7634 


7904 


8173 


8441 


8710 


8979 


9247 




162 


9515 


9783 


..61 


.319 


.586 


.853 


1121 


1388 


1654 


1921 




163 


212188 


2454 


2720 


2986 


3252 


3518 


3783 


4049 


4314 


4579 




164 


4844 


5109 


5373 


5638 


5902 


6166 


6430 


6694 


6957 


7221 




165 


7484 


7747 


8010 


8273 


8536 


8798 


9060 


9323 


9585 


9846 




166 


220108 


0370 


0631 


0892 


1153 


1414 


1675 


1936 


2196 


2456 




167 


2716 


2976 


3236 


3496 


3755 


4015 


4274 


4533 


4792 


6051 




168 

.I 169 


5309 


5568 


5S26 


6084 


6342 


6600 


6868 


7115 


7372 


7630 




7887 


8144 


8400 


8667 


8913 


9170 


9426 


9682 


9938 


.193 




170 


230449 


0704 


0960 


1216 


1470 


1724 


1979 


2234 


2488 


2742 




171 


2996 


3250 


3504 


3757 


4011 


4264 


4517 


4770 


5023 


5276 




172 


5528 


5781 


6033 


6285 


6537 


6789 


7041 


7292 


7644 


7795 




173 


8046 


8297 


8548 


8799 


9049 


9299 


9550 


9800 


..50 


.300 




174 


240549 


0799 


1048 


1297 


1546 


1795 


2044 


2293 


2541 


2790 




175 


3038 


3288 


3534 


3782 


4030 


4277 


4525 


4772 


5019 


6266 




176 


5513 


5759 


6006 


6252 


6499 


6745 


6991 


7237 


7482 


7728 




177 


7973 


8219 


8464 


8709 


8954 


9198 


9443 


9687 


9932 


.176 




178 


250420 


0664 


0908 


1151 


1395 


1638 


1881 


2125 


2368 


2610 




179 


2853 


3096 


3338 


3580 


3822 


4064 


4306 


4548 


4790 


6031 




180 


5273 


5514 


5755 


5996 


6237 


6477 


6718 


6958 


7198 


7439 




181 


7679 


7918 


8158 


8398 


8637 


8877 


9116 


9355 


9594 


9833 




182 


260071 


0310 


0548 


0787 


1025 


1263 


1501 


1739 


1976 


2214 




183 


2451 


2688 


2925 


3162 


3399 


3836 


3873 


4109 


4346 


4582 




184 


4818 


5054 


5290 


5525 


6761 


5996 


6232 


6467 


6702 


6937 




185 


7172 


7406 


7641 


7875 


8110 


8344 


8578 


8812 


9046 


9279 




186 


9513 


9746 


9980 


.213 


.446 


.679 


.912 


1144 


1377 


1609 




187 


271842 


2074 


2306 


2538 


2770 


3001 


3233 


3464 


3696 


3927 




188 


4158 


4389 


4620 


4850 


5081 


5311 


5542 


5772 


6002 


6232 




189 


6462 


6692 


6921 


7151 


7380 


7609 


7838 


8067 


8296 


8525 




190 


8754 


8982 


9211 


9439 


9667 


9895 


.123 


.351 


.578 


.806 




191 


281033 


1261 


1488 


1715 


1942 


2169 


2396 


2622 


2849 


3075 




192 


3301 


3527 


3753 


3979 


4205 


4431 


4656 


!"4882 


5107 


5332 




193 


6557 


5782 


6007 


6232 


6456 


6681 


6905 


7130 


7354 


7578 




194 


7802 


8026 


8249 


8473 


8696 


8920 


9143 


9366 


9589 


9812 




195 


290035 


0257 


0480 


0702 


0925 


1147 


1369 


1591/ 


1813 


2034 




196 


2258 


2478 


2699 


2920 


3141 


3363 


3684 


3804 


4025 


4246 




197 


4466 


4687 


4907 


5127 


6347 


5567 


5787 


6007 


6226 


6446 




198 


6665 


6884 


7104 


7323 


7542 


7761 


7979 


8198 


8416 


8635 




199 


8853 


9071 


9289 


9507 


9725 


9943 


.161 


.378 


.595 


.813 





OF NUMBERS 5 




N. 





1 


2 


3 


4 | 5 


6 


7 


S 


9 




200 


301030 


1247 


1464 


1681 


1898 


2114 


2331 


2547 


2764 


2980 




201 


3196 


3412 


3628 


3844 


4059 


4275 


4491 


4706 


4921 


5136 




202 


5351 


5566 


5781 


5996 


6211 


6425 


6639 


6854 


7068 


7282 


1 


203 


7496 


7710 


7924 


8137 


8351 


8564 


8778 


8991 


9204 


9417 




204 


9630 


9843 


..56 


.268 


.481 


.693 


.906 


1118 


1330 


1542 




205 


311754 


1966 


2177 


2389 


2600 


2812 


3023 


3234 


3445 


3656 




206 


3867 


4078 


4289 


4499 


4710 


4920 


5130 


5340 


5551 


5760 




207 


5970 


6180 


6390 


6599 


6809 


7018 


7227 


7436 


7646 


7854 




208 


8063 


8272 


8481 


8689 


8898 


9106 


9314 


9522 


9730 


9938 




209 


320146 


0354 


0562 


0769 


0977 


1184 


1391 


1598 


1805 


2012 




210 


2219 


2426 


2633 


2839 


3046 


3252 


3458 


3665 


3871 


4077 




211 


4282 


4488 


4694 


4899 


5105 


5310 


5516 


5721 


5926 


6131 




212 


6336 


6541 


6745 


6950 


7155 


7359 


7563 


7767 


7972 


8176 




213 


8380 


8583 


8787 


8991 


9194 


9398 


9601 


9805 


...8 


.211 




214 


330414 


0617 


0819 


1022 


1225 


1427 


1630 


1832 


2034 


2236 




215 


2438 


2640 


2842 


3044 


3246 


3447 


3649 


3850 


4051 


4253 




216 


4454 


4655 


4856 


5057 


5257 


5458 


5658 


5859 


6059 


6260 




217 


6460 


6660 


6860 


7060 


7260 


7459 


7659 


7858 


8058 


8257 




218 


8456 


8656 


8855 


9054 


9253 


9451 


9650 


9849 


. .47 


.246 




219 


340444 


0642 


0841 


1039 


1237 


1435 


1632 


1830 


2028 


2225 




220 


2423 


2620 


2817 


3014 


3212 


3409 


3606 


3802 


3999 


4196 




221 


4392 


4589 


4785 


4981 


5178 


5374 


5570 


5766 


5962 


6157 




222 


6353 


6549 


6744 


6939 


7135 


7330 


7525 


7720 


7915 


8110 




223 


8305 


8500 


8694 


8889 


9083 


9278 


9472 


9666 


9860 


. .54 




224 


350248 


0442 


0636 


0829 


1023 


1216 


1410 


1603 


1796 


1989 




225 


2183 


2375 


2568 


2761 


2954 


3147 


3339 


3532 


3724 


3916 




226 


4108 


4301 


4493 


4685 


4876 


5068 


5260 


5452 


5643 


5834 




227 


6026 


6217 


6408 


6599 


6790 


6981 


7172 


7363 


7554 


7744 




228 


7935 


8125 


8316 


8506 


8696 


8886 


9076 


9266 


9456 


9646 




229 


9835 


..25 


.215 


.404 


.593 


.783 


.972 


1161 


1350 


1539 




230 


361728 


1917 


2105 


2294 


2482 


2671 


2859 


3048 


3236 


8424 




231 


3612 


3800 


3988 


4176 


4363 


4551 


4739 


4926 


5113 


5301 




232 


5488 


5675 


5862 


6049 


6236 


6423 


6610 


6796 


6983 


7169 




233 


7356 


7542 


7729 


7915 


8101 


8287 


8473 


8659 


8845 


9030 




234 


9216 


9401 


9587 


9772 


9958 


.143 


.328 


.513 


.698 


.883 




235 


371068 


1253 


1437 


1622 


1806 


1991 


2175 


2360 


2544 


2728 




236 


2912 


3096 


3280 


3464 


3647 


3831 


4015 


4198 


4382 


4565 




237 


4748 


4932 


5115 


5298 


5481 


6664 


5846 


6029 


6212 


6394 




238 


6577 


6759 


6942 


7124 


7306 


7488 


7670 


7852 


8034 


8216 




239 


8398 


8580 


8761 


8943 


9124 


9306 


9487 


9608 


9849 j . .30 


j 


240 


380211 


0392 


0573 


0754 


0934 


1115 


1296 


1476 


1656 1837 


\ 


241 


2017 


2197 


2377 


2557 


2737 


2917 


3097 


3277 


3456 3636 


\ 


242 


3815 


3995 


4174 


4353 


4533 


4712 


4891 


5070 


5249 5428 




243 


5606 


5785 


5964 


6142 


6321 


6499 


6677 


6856 


7034 7212 


1 


244 


7390 


7568 


7746 


7923 


8101 


8279 


8456 


8634 


8811 8989 


I 


245 


9166 


9343 


9520 


9698 


987o 


..51 


.228 


.405 


.582 ' .759 


\ 


246 


390935 


1112 


1288 


1464 


1641 


1817 


1993 


2169 


2345 2521 




247 


2697 


2873 


3048 


3224 


3400 


3575 


3751 


3926 


4101 4277 




248 


4452 


4627 


4802 


4977 


5152 


5326 


5501 


6676 


5850 6025 




249 


6199 


6374 


6548 


6722 


6896 1 7071 


7245 


7419 


7592 7766 







6 


LOGARITHMS 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




250 


397940 


8114 


8287 


8461 


8634 


8808 


8981 


9154 


9328 


9501 




251 


9674 


9847 


..20 


.192 


.365 


.638 


.711 


.883 


1056 


1228 




252 


401401 


1573 


1745 


1917 


2089 


2261 


2433 


2605 


2777 


2949 




253 


3121 


3292 


3464 


3635 


3807 


3978 


4149 


4320 


4492 


4663 




254 


4834 


6005 


5176 


5346 


5517 


5688 


5858 


6029 


6199 


6370 




255 


" 6540 


6710 


6881 


7051 


7221 


7391. 


7561 


7731 


7901 


8070 




256 


8240 


8410 


8579 


8749 


8918 


9087 


9257 


9426 


9595 


9764 




257 


9933 


.102 


.271 


.440 


.609 


.777 


.946 


1114 


1283 


1451 




258 


411620 


1788 


1956 


2124 


2293 


2461 


2629 


2796 


2964 


3132 




259 


3300 


3467 


3635 


3803 


3970 


4137 


4305 


4472 


4639 


4806 




260 


4973 


5140 


5307 


5474 


5641 


5808 


5974 


6141 


6308 


6474 




261 


6641 


6807 


6973 


7139 


7306 


7472 


7638 


7804 


7970 


8135 




262 


8301 


S467 


8633 


8798 


8964 


9129 


9295 


9460 


9625 


9791 




263 


9956 


.121 


.286 


.451 


.616 


.781 


.945 


1110 


1275 


1439 




264 


421604 


1788 


1933 


2097 


2261 


2426 


2590 


2754 


2918 


3082 




265 


3246 


3410 


3574 


3737 


3901 


4065 


4228 


4392 


4555 


4718 




266 


4882 


5045 


5208 


5371 


5534 


5697 


5860 


6023 


6186 


6349 




267 


6511 


6674 


6836 


6999 


7161 


7324 


7486 


7648 


7811 


7973 




268 


8135 


8297 


8459 


8621 


8783 


8944 


9106 


9268 


9429 


9591 




269 


9752 


9914 


..75 


.236 


.398 


.559 


.720 


.881 


1042 


1203 




270 


431364 


1525 


1685 


1846 


2007 


2167 


2328 


2488 


2649 


2809 




271 


2969 


3130 


3290 


3450 


3610 


3770 


3930 


4090 


4249 


4409 




272 


4569 


4729 


4888 


5048 


5207 


5367 


5526 


5685 


5844 


6004 




273 


6163 


6322 


6481 


6640 


6800 


6957 


7116 


7275 


7433 


7692 




274 


7751 


7909 


8067 


8226 


8384 


8542 


8701 


8859 


9017 


9175 




275 


9333 


9491 


9648 


9806 


9964 


.122 


.279 


.437 


.594 


.752 




276 


440909 


1066 


1224 


1381 


1538 


1695 


1852 


2009 


2166 


1323 




277 


2480 


2637 


2793 


2950 


3106 


3263 


3419 


3576 


3732 


3889 




278 


4045 


4201 


4357 


4513 


4669 


4825 


4981 


5137 


5293 


6449 




279 


5604 


5760 


5915 


6071 


6226 


6382 


6537 


6692 


6848 


7003 




280 


7158 


7313 


7468 


7623 


'7778 


7933 


8088 


8242 


8397 


8552 




281 


8706 


8861 


9015 


9170 


9324 


9478 


9633 


9787 


9941 


. .95 




282 


450249 


0403 


0557 


0711 


0865 


1018 


1172 


1326 


1479 


1633 




283 


1786 


1940 


2093 


2247 


2400 


2553 


2706 


2859 


3012 


3165 




284 


3318 


3471 


3624 


3777 


3930 


4082 


4235 


4387 


4540 


4692 




2S5 


4845 


4997 


5150 


5302 


5454 


5606 


5758 


5910 


6062 


6214 




286 


6366 


6518 


6670 


6821 


6973 


7125 


7276 


7428 


7579 


7731 




287 


7882 


8033 


8184 


8336 


8487 


8638 


8789 


8940 


9091 


9242 




288 


9392 


9543 


9694 


9845 


9995 


.146 


.296 


.447 


.597 


.7-18 




289 


460898 


1048 


1198 


1348 


1499 


1649 


1799 


1948 


2098 


2248 




290 


2398 


2548 


2697 


2847 


2997 


3146 


3296 


3445 


3594 


3744 




291 


3893 


4042 


4191 


4340 


4490 


4639 


4788 


4936 


5085 


5234 




292 


5383 


5532 


5680 


5829 


5977 


6126 


6274 


6423 


6571 


6719 




293 


6868 


7016 


7164 


7312 


7460 


7608 


7756 


7904 


8052 


8200 




294 


8347 


8495 


8643 


8790 


8938 


9085 


9233 


9380 


9527 


9675 




295 


9822 


9969 


.116 


.263 


.410 


.557 


.704 


.851 


.998 


1145 




296 


471292 


1438 


1585 


1732 


1878 


2025 


2171 


2318 


2464 


2610 




297 


2756 


•2903 


3049 


3195 


3341 


3487 


3633 


3779 


S925 


4071 




298 


4216 


4362 


4508 


4653 


4799 


4944 


5090 


5235 


6381 


5526 




299 


5671 


5816 


5962 


6107 


6252 


6397 


6542 


6687 


6832 


6976 



OF NUMBERS. 7 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


300 


477121 


7266 


7411 


7555 


7700 


7844 


7989 


8133 


8278 


8422 


301 


8566 


8711 


8855 


8999 


9143 


9287 


9481 


9575 


9719 


9863 


302 


480007 


0151 


0294 


0438 


0582 


0725 


0869 


1012 


1156 


1299 


303 


1443 


1586 


1729 


1872 


2016 


2159 


2302 


2445 


2588 


2731 


304 


2874 


3016 


3159 


3302 


3445 


3587 


3730 


3872 


4015 


4157 


305 


4300 


4442 


4585 


4727 


4869 


5011 


5153 


5295 


6437 


5579 


306 


5721 


6863 


6005 


6147 


6289 


6430 


6572 


6714 


6855 


6997 


307 


7138 


7280 


7421 


7663 


7704 


7845 


7986 


8127 


8269 


8410 


308 


8551 


8692 


8833 


8974 


9114 


9255 


9396 


9537 


9667 


9818 


309 


9959 


..99 


.239 


.380 


.520 


.661 


.801 


.941 


1081 


1222 


310 


491362 


1502 


1642 


1782 


1922 


2062 


2201 


2341 


2481 


2621 


311 


2760 


2900 


3040 


3179 


3319 


3458 


3597 


3737 


3876 


4015 


312 


4155 


4294 


4433 


4572 


4711 


4850 


4989 


6128 


5267 


5406 


313 


6544 


5683 


5822 


5960 


6099 


6238 


6376 


6515 


6653 


6791 


314 


6930 


7068 


7206 


7344 


7483 


7621 


7759 


7897 


8035 


8173 


315 


8311 


8448 


8586 


8724 


8862 


8999 


9137 


9275 


9412 


9550 


316 


9687 


9824 


9962 


..99 


.236 


.374 


.511 


.648 


.785 


.922 


317 


501059 


1196 


1333 


1470 


1607 


1744 


1880 


2017 


2154 


2291 


318 


2427 


2564 


2700 


2837 


2973 


3109 


3246 


3382 


3518 


3655 


319 


3791 


3927 


4063 


4199 


4335 


4471 


4607 


4743 


1878 


6014 


320 


5150 


5286 


5421 


5557 


5693 


5828 


5964 


6099 


6234 


6370 


321 


6505 


6640 


6776 


6911 


7046 


7181 


7316 


7451 


7586 


7721 


322 


7856 


7991 


812S 


8260 


8395 


8630 


8664 


8799 


8934 


9008 


323 


9203 


9337 


9471 


9606 


9740 


9874 


...9 


.143 


.277 


.411 


324 


510545 


0679 


0813 


0947 


1081 


1215 


1349 


1482 


1616 


1750 


325 


1883 


2017 


2151 


2284 


2418 


2551 


2684 


2818 


2951 


3084 


326 


3218 


3351 


3484 


3617 


3750 


3883 


4016 


4149 


4282 


4414 


327 


4548 


4681 


4813 


4946 


5079 


6211 


5344 


5476 


5609 


5741 


328 


5874 


6006 


6139 


6271 


6403 


6535 


6668 


6800 


6932 


7064 


329 


7196 


7328 


7460 


7592 


7724 


7855 


7987 


8119 


8251 


8382 


330 


8514 


8646 


8777 


8909 


9040 


9171 


9303 


9434 


9566 


9697 


331 


9828 


9959 


..90 


.221 


.353 


.484 


.615 


.745 


.876 


1007 


332 


521138 


1269 


1400 


1530 


1661 


1792 


1922 


2053 


2183 


2314 


333 


2444 


2575 


2705 


2835 


2966 


3096 


3226 


3356 


3486 


3616 


334 


3746 


3876 


4006 


4136 


4266 


4396 


4526 


4656 


4785 


4915 


335 


5045 


5174 


5304 


5434 


6563 


5693 


6822 


5951 


6081 


6210 


336 


6339 


6469 


6598 


6727 


6856 


6985 


7114 


7243 


7372 


7501 


337 


7630 


7759 


7888 


8016 


8145 


8274 


8402 


8531 


8660 


8788 


338 


8917 


9045 


9174 


9302 


9430 


9559 


9687 


9815 


9943 


..72 


339 


530200 


0328 


0456 


0584 


0712 


0840 


0968 


1096 


1223 


1S51 


340 


1479 


1607 


1734 


1862 


1960 


2117 


2245 


2372 


2500 


2627 


341 


2754 


2882 


3009 


3136 


3264 


3391 


3518 


3645 


3772 


3899 


342 


4026 


4153 


4280 


4407 


4534 


4661 


4787 


4914 


5041 


5167 


343 


5294 


6421 


5547 


5674 


5800 


5927 


6053 


6180 


6306 


6432 


344 


6658 


6685 


6811 


6937 


7060 


7189 


7315 


7441 


/567 


7693 


345 


7819 


7945 


8071 


8197 


8322 


8448 


8574 


8699 


8825 


8951 


346 


9076 


9202 


9327 


9452 


9578 


9703 


9829 


9954 


..79 


.204 


347 


540329 


0455 


0580 


0705 


0830 


0955 


1080 


1205 


1330 


1454 


348 


1579 


1704 


1829 


1953 


2078 


2203 


2327 


2452 


2576 


2701 


349 


2825 


2950 


3074 


3199 


3323 


3447 


3571 


3696 


3820 


3944 



8 


LOGARITHMS 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




360 


544068 


4192 


4316 


4440 


4564 


4688 


4812 


4936 


5060 


5183 




1 351 


5307 


5431 


6555 


5678 


6805 


6925 


6049 


6172 


6296 


6419 




"352 


6543 


6666 


6789 


6913 


7036 


7159 


7282 


7405 


7529 


7652 




353 


7775 


7898 


8021 


8144 


8267 


8389 


8512 


8635 


8758 


8881 




• 354 


9003 


9126 


9249 


9371 


9494 


9616 


9739 


9861 


9984 


.196 




355 


550228 


0351 


0473 


0595 


0717 


0840 


0962 


1084 


1206 


1328 




356 


1450 


1572 


1694 


1816 


1938 


2060 


2181 


2303 


2425 


2547 




357 


2668 


2790 


2911 


3033 


3155 


3276 


3393 


3519 


3640 


3762 




358 


3883 


4004 


4126 


4247 


4368 


4489 


4610 


4731 


4852 


4973 




359 


5094 


5215 


6346 


6457 


6578 


5699 


6820 


5940 


6061 


6182 




360 


6303 


6423 


6544 


6664 


6785 


6905 


7026 


7146 


7267 


7387 




361 


7507 


7627 


7748 


7868 


7988 


8108 


8228 


8349 


8469 


8589 




362 


8709 


8829 


8948 


9068 


9188 


9308 


9428 


9548 


9667 


9787 




363 


9907 


. .26 


.146 


.265 


.385 


.604 


.624 


.743 


.863 


.982 




! 364 


561101 


1£21 


1340 


1459 


1578 


1698 


1817 


1936 


2055 


2173 




365 


2293 


2412 


2531 


2650 


2769 


2887 


3006 


3125 


3244 


3362 




366 


3481 


3600 


3718 


3837 


3955 


4074 


4192 


4311 


4429 


4548 




367 


4666 


4784 


4903 


5021 


5139 


6257 


5376 


5494 


5612 


6730 




368 


6848 


5966 


6084 


6202 


6320 


6437 


6655 


6673 


6791 


6909 




369 


7026 


7144 


7262 


7379 


7497 


7614 


7732 


7849 


7967 


8084 




370 


8202 


8319 


8436 


8554 


8671 


8788 


8905 


9023 


9140 


9257 




371 


9374 


9491 


9608 


9725 


9882 


9959 


..76 


.193 


.309 


.426 




i 372 


570543 


0660 


0776 


0893 


1010 


1126 


1243 


1359 


1476 


1592 




373 


1709 


1825 


1942 


2058 


2174 


2291 


2407 


2523 


2639 


2755 




374 


2872 


2988 


3104 


3220 


3336 


3452 


3568 


3684 


3800 


£915 




375 


4031 


4147 


4263 


4379 


4494 


4610 


4726 


4841 


4957 


5072 




376 


6188 


5303 


6419 


5534 


5650 


5766 


5880 


5996 


6111 


6226 




i 377 


6341 


6457 


6572 


6687 


6802 


6917 


7032 


7147 


7262 


7377 




378 


7492 


7607 


7722 


7836 


7951 


8066 


8181 


8295 


8410 


8525 




379 


8639 


8754 


8868 


8983 


9097 


9212 


9326 


9441 


9555 


9669 




380 


9784 


9898 


..12 


.126" 


.241 


.355 


.469 


.583 


.697 


.811 




381 


580925 


1039 


1153 


1267 


1381 


1495 


1608 


1722 


1836 


196t 




382 


2063 


2177 


2291 


2404 


2518 


2631 


2745 


2858 


2972 


3085 




383 


3199 


3312 


3426 


3539 


3652 


3765 


3879 


3992 


4105 


4218 




384 


4331 


4444 


4557 


4670 


4783 


4896 


5009 


5122 


5235 


5348 




385 


5461 


5574 


6686 


5799 


6912 


6024 


6137 


6250 


6362 


6475 




386 


6587 


6700 


6812 


6925 


7037 


7149 


7262 


7374 


7486 


7599 




387 


7711 


7823 


7935 


8047 


8160 


8272 


8384 


8496 


8608 


8720 




388 


8832 


8944 


9056 


9167 


9279 


9391 


9503 


9615 


9726 


9834 




389 


9950 


..61 


.173 


.284 


.396 


.607 


.619 


.730 


.842 


.963 




390 


691065 


1176 


1287 


1399 


1510 


1621 


1732 


1843 


1955 


2066 




391 


2177 


2288 


2399 


2510 


2621 


2732 


2843 


2954 


3064 


3175 




392 


3286 


3397 


3508 


3618 


3729 


3840 


3950 


4061 


4171 


4282 




393 


4393 


4503 


4614 


4724 


4834 


4945 


6055 


5165 


5276 


5386 




394 


5496 


6606 


6717 


5827 


6937 


6047 


6167 


6267 


6377 


6487 




395 


6597 


6707 


6817 


6927 


7037 


7146 


7256 


7366 


7476 


7586 




396 


7695 


7805 


7914 


8024 


8134 


8243 


8353 


8462 


8572 


8681 




397 


8791 


8900 


9009 


9119 


9228 


9337 


9446 


9556 


9666 


9774 




398 


9883 


9992 


.101 


.210 


.319 


.428 


.537 


.646 


.755 


.864 




399 


600973 


1082 


1191 


1299 


1408 


1617 


1625 


1734 


1843 


1951 







F NUMBERS 






9 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




400 


602060 


2169 


2277 


2386 


2494 


2603 


2711 


2819 


2928 


3036 




401 


3144 


3253 


3361 


3469 


3573 


3686 


3794 


3902 


4010 


4118 




402 


4226 


4334 


4442 


4550 


4658 


4766 


4874 


4982 


5089 


5197 




403 


5305 


5413 


5521 


5628 


5736 


5844 


5951 


6059 


6166 


6274 




404 


6381 


6489 


6596 


6704 


6811 


6919 


7026 


7133 


7241 


7348 




405 


7455 


7562 


7669 


7777 


7884 


7991 


8098 


8205 


8312 


8419 




406 


8526 


8633 


8740 


8847 


8954 


9061 


9167 


9274 


9381 


9488 




407 


9594 


9701 


9808 


9914 


..21 


.128 


.234 


.341 


.447 


.554 




408 


610660 


0767 


0873 


0979 


1086 


1192 


1298 


1405 


1511 


1617 




409 


1723 


1829 


1936 


2042 


2148 


2254 


2360 


2466 


2572 


2678 




410 


2784 


2890 


2996 


3102 


3207 


3313 


3419 


3525 


3630 


3736 




411 


3842 


3947 


4053 


4159 


4264 


4370 


4476 


4581 


4686 


4792 




412 


4897 


5003 


5108 


5213 


5319 


5424 


6529 


5634 


5740 


6845 




413 


5950 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


6790 


6895 




414 


7000 


7105 


7210 


7315 


7420 


7525 


7629 


7734 


7839 


7943 




415 


8048 


8153 


8257 


8362 


8466 


8571 


8676 


8780 


8884 


8989 




416 


9293 


9198 


9302 


9406 


9511 


9615 


9719 


9824 


9928 


..32 




417 


620136 


0140 


0344 


0448 


0552 


0656 


0760 


0864 


0968 


1072 




418 


1176 


1280 


1384 


1488 


1592 


1695 


1799 


1903 


2007 


2110 




419 


2214 


2318 


2421 


2525 


2628 


2732 


2835 


2939 


3042 


3146 




420 


3249 


3353 


3456 


3559 


3663 


3766 


3869 


3973 


4076 


4179 




421 


4282 


4385 


4488 


4591 


4695 


4798 


4901 


5004 


6107 


5210 




422 


5312 


5415 


5518 


5621 


6724 


5827 


5929 


6032 


6135 


6238 




423 


6340 


6443 


6546 


6648 


6751 


6853 


6956 


7058 


7161 


7263 




424 


7366 


7468 


7571 


7673 


7775 


7878 


7980 


8082 


8185 


8287 




425 


8389 


8491 


8593 


8695 


8797 


8900 


9002 


9104 


9206 


9308 




426 


9410 


9512 


9613 


9715 


9817 


9919 


..21 


.123 


.224 


.326 




427 


630428 


0530 


0631 


0733 


0835 


0936 


1038 


1139 


1241 


1342 




428 


1444 


1545 


1647 


1748 


1849 


1951 


2052 


2153 


2255 


2356 




429 


2457 


2559 


2660 


2761 


2862 


2963 


3064 


3165 


3266 


3367 




430 


3468 


3569 


3670 


3771 


3872 


3973 


4074 


4175 


4276 


4376 




431 


4477 


4578 


4679 


4779 


4880 


4981 


5081 


5182 


5283 


5383 




432 


5484 


5584 


5685 


5785 


5886 


5986 


6087 


6187 


6287 


6388 




433 


6488 


6588 


6688 


6789 


6889 


6989 


7089 


7189 


7290 


7390 




434 


7490 


7590 


7690 


7790 


7890 


7990 


8090 


8190 


8290 


8389 




435 


8489 


8589 


8689 


8789 


8888 


8988 


9088 


9188 


9287 


9387 




436 


9486 


9586 


9686 


9785 


9885 


9984 


..84 


.183 


.283 


.382 




437 


640481 


0581 


0680 


0779 


0879 


0978 


1077 


1177 


1276 


1375 




438 


1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2366 




439 


2465 


2563 


2662 


2761 


2860 


2959 


3058 


3156 


3255 


3354 




440 


3453 


3551 


3650 


3749 


3847 


3946 


4044 


4143 


4242 


4340 




441 


4439 


4537 


4636 


4734 


4832 


4931 


5029 


6127 


5226 


5324 




442 


5422 


5521 


5619 


5717 


5815 


6913 


6011 


6110 


6208 


6306 




443 


6404 


6502 


6600 


6698 


6796 


6894 


6992 


7089 


7187 


7285 




444 


7383 


7481 


7579 


7676 


7774 


7872 


7969 


8067 


8165 


8262 




445 


8360 


8458 


8555 


8653 


8750 


8848 


8945 


9043 


9140 


9237 




446 


9335 


9432 


9530 


9627 


9724 


9821 


9919 


..16 


.113 


.210 




447 


650808 


0405 


0502 


0599 


0696 


0793 


0890 


0987 


1084 


1181 




1 448 


1278 


1375 


1472 


1569 


1666 


1762 


1859 


1956 


2053 


2150 




! 449 


2246 


2343 


2440 


2530 


2633 


2730 


2826 


2923 


3019 


3116 





|:o 




LOGARITHMS 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


450 


653213 


3309 


3405 


3502 


3598 


3695 


3791 


3888 


3984 


4080 


451 


4177 


4273 


4369 


4465 


4562 


4658 


4754 


4850 


4946 


5042 


452 


5138 


5235 


5331 


5427 


5526 


5619 


5715 


5810 


5906 


6002 


453 


6098 


6194 


6290 


6386 


6482 


6577 


6673 


6769 


6864 


6960 


454 


7056 


7152 


7247 


7343 


7438 


7534 


7629 


7725 


7820 


7916 


455 


8011 


8107 


8202 


8298 


8393 


8488 


8584 


8679 


8774 


8870 


456 


8965 


9U60 


S155 


9250 


9346 


9441 


9536 


9631 


9726 


9821 


457 


9916 


. .11 


.106 


.201 


.296 


.391 


.486 


.581 


.676 


.771 


458 


660865 


0960 


1055 


1150 


1245 


1339 


1434 


1529 


1623 


1718 


459 


1813 


190/ 


2002 


2096 


2191 


2286 


2380 


2475 


2569 


2663 


460 


2758 


2852 


2947 


3041 


3135 


3230 


3324 


3418 


3512 


S607 


461 


3701 


3795 


3889 


3983 


4078 


4172 


4266 


4360 


4454 


4548 


462 


4642 


4736 


48o0 


4924 


5018 


5112 


5206 


5299 


5393 


5487 


463 


5581 


5675 


57G9 


6862 


5956 


6050 


6143 


6237 


6331 


6424 


464 


6518 


6612 


6705 


6799 


6892 


6986 


7079 


7173 


7266 


7360 


465 


7453 


7546 


7640 


7733 


7826 


7920 


8013 


8106 


8199 


8293 


466 


8386 


8479 


8572 


8665 


8759 


8852 


8945 


9038 


9131 


9324 


467 


9317 


9410 


9503 


9596 


9689 


9782 


9875 


9967 


..60 


.153 


468 


670241 


0339 


0431 


0524 


0617 


0710 


0802 


0895 


0988 


1080 


469 


1173 


1265 


1358 


1451 


1543 


1636 


1728 


1821 


1913 


2005 


470 


2098 


2190 


2283 


2375 


2467 


2560 


2652 


2744 


2836 


2929 


471 


3021 


3113 


3205 


3297 


3390 


3482 


3574 


3666 


3758 


3850 


472 


3942 


4034 


4126 


4218 


4310 


4402 


4494 


4586 


4677 


4769 


473 


4861 


4953 


5045 


5137 


5228 


5320 


5412 


6503 


5595 


5687 


474 


5778 


5870 


5962 


6053 


6145 


6236 


6328 


6419 


6511 


6b02 


475 


6694 


6785 


6876 


6968 


7059 


7151 


7242 


7333 


7424 


7516 


476 


7607 


7698 


7789 


7881 


7972 


8063 


8154 


8245 


8336 


8427 


477 


8518 


8609 


8700 


8791 


8882 


8972 


9064 


9155 


9246 


9337 


478 


9428 


9519 


9610 


9700 


9791 


9882 


9973 


..63 


.154 


.245 


479 


680336 


0426 


0517 


0607 


0698 


0789 


0879 


0970 


1060 


1151 


480 


1241 


1332 


1422 


1513* 


1603 


1693 


1784 


1874 


1964 


2055 


481 


2145 


2235 


2326 


2416 


2506 


2596 


2686 


2777 


2867 


2957 


482 


3047 


3137 


3227 


3317 


3407 


3497 


3587 


3677 


3767 


3857 


483 


3947 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4756 


484 


4854 


4935 


5025 


5114 


5204 


5294 


5383 


5473 


5563 


5652 


485 


5742 


5831 


5921 


6010 


6100 


6189 


6279 


6368 


6458 


6547 


486 


6636 


6726 


6815 


6904 


6994 


7083 


7172 


7261 


7351 


7440 


487 


7529 


7618 


7707 


7796 


7886 


7975 


8064 


8153 


8242 


8331 


488 


8420 


8509 


8598 


8687 


8776 


8865 


8953 


9042 


9131 


9220 


489 


9309 


9398 


9486 


9575 


9664 


9753 


9841 


9930 


..19 


.107 


490 


690196 


0285 


0373 


0362 


0550 


0639 


0728 


0816 


0905 


0993 


491 


1081 


1170 


1258 


1347 


1435 


1524 


1612 


1700 


1789 


1877 


492 


1965 


2053 


2142 


2230 


2318 


2408 


2494 


2583 


2671 


2759 


493 


2847 


2935 


3023 


3111 


3199 


3287 


3375 


3463 


3551 


3639 


\ 494 


3727 


3815 


3903 


3991 


4078 


4166 


4254 


4342 


4430 


4517 


495 


4605 


4893 


4781 


4868 


4956 


5044 


5131 


5210 


5307 


5394 


496 


5482 


55(39 


5657 


5744 


5832 


5919 


6007 


6094 


6182 


6269 


497 


6356 


5444 


6531 


6618 


6706 


6793 


6880 


6988 


7055 


7142 


498 


7229 


7317 


7404 


7491 


7578 


7665 


7752 


7839 


7926 


8014 


499 


8101 


8188 


8276 


8362 


8449 


8535 


8622 


8709 


8796 


8883 







OF NUMBERS. 






11 




N. 





I 


2 


3 


4. 


5 


6 


7 


8 


9 




500 


698970 


9057 


9144 


9231 


9317 


9404 


9491 


9578 


9664 


9751 




501 


9838 


9924 


. .11 


..98 


.184 


.271 


.358 


.444 


.531 


.617 




502 


700704 


0790 


0877 


0963 


1050 


1136 


1222 


1309 


1395 


1482 




503 


1568 


1654 


1741 


lb27 


1913 


1999 


2086 


2172 


2258 


2344 




504 


2431 


2517 


2603 


2689 


2775 


2861 


2947 


3033 


3119 


3205 




505 


3291 


3377 


3463 


3549 


3635 


3721 


3807 


3895 


3979 


4065 




503 


4151 


4236 


4322 


4408 


4494 


4579 


4665 


4751 


4837 


4922 




507 


5008 


5094 


5179 


5265 


5350 


5436 


5522 


5607 


5693 


5778 




503 


5864 


5949 


6035 


6120 


6206 


6291 


6376 


6462 


6547 


6632 




509 


6718 


6803 


6888 


6974 


7059 


7144 


7229 


7315 


7400 


7485 




510 


7570 


7655 


7740 


7826 


7910 


7996 


8081 


8166 


8251 


8336 




511 


8421 


8506 


8591 


8676 


8761 


8846 


8931 


9015 


9100 


9185 




612 


9270 


9355 


9440 


9524 


9609 


S694 


9779 


9863 


9948 


. .33 




513 


710117 


0202 


0287 


0371 


0456 


0540 


0625 


0710 


0794 


0879 




514 


0963 


1048 


1132 


1217 


1301 


1385 


1470 


1554 


1639 


1723 




515 


1807 


1892 


1976 


2030 


2144 


2229 


2313 


2397 


2481 


2566 




516 


2650 


2734 


2818 


2902 


2986 


3070 


3154 


3238 


3326 


3407 




517 


3491 


3575 


3659 


3742 


3826 


3910 


3994 


4078 


4162 


4243 




618 


4330 


4414 


4497 


4581 


4665 


4749 


4833 


4916 


5000 


5034 


j 


519 


6167 


5251 


5335 


5418 


5502 


6586 


5o69 


5753 


5836 


5920 




520 


6003 


6087 


6170 


6254 


6337 


6421 


6504 


6588 


6671 


6754 




521 


6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


7504 


7587 


| 


522 


7671 


7754 


7837 


7920 


8003 


8086 


8169 


8253 


8336 


8419 




523 


8502 


8585 


8668 


8751 


8834 


8917 


9000 


9083 


9165 


9248 




524 


9331 


9414 


9497 


9580 


9663 


9745 


9828 


9911 


9994 


..77 




525 


720159 


0242 


0325 


0407 


0490 


0573 


0655 


0738 


0821 


0903 




526 


0986 


1068 


1151 


1233 


1316 


1398 


1481 


1563 


1646 


172S 




527 


1811 


1893 


.975 


2058 


2140 


2222 


2305 


2387 


2469 


2552 




528 


2634 


2716 


2798 


2881 


2963 


3045 


3127 


3209 


3291 


3374 




529 


3456 


3538 


3620 


3702 


3784 


3866 


3948 


4030 


4112 


4194 




630 


4276 


4358 


4440 


4522 


4604 


4685 


4767 


4849 


4931 


5013 




531 


5095 


6176 


5258 


5340 


5422 


5503 


5585 


5667 


5748 


5830 




532 


6912 


6993 


6075 


6156 


6238 


6320 


6401 


6483 


6564 


6646 




533 


6727 


6809 


6890 


6972 


7053 


7134 


7216 


7297 


7379 


7460 




' 634 


7541 


7623 


7704 


7785 


7866 


7948 


8029 


8110 


8191 


8273 




535 


8354 


8435 


8516 


8597 


8678 


8759 


8841 


8922 


9003 


9084 




536 


9165 


9246 


9327 


9403 


9489 


9570 


9651 


9732 


9813 


9893 




537 


9974 


..65 


.136 


.217 


.298 


.378 


.459 


.440 


.621 


.702 




638 


730782 


0863 


0944 


1024 


1105 


1186 


1266 


1347 


1428 


1508 




539 


1589 


1669 


1750 


1830 


1911 


1991 


2072 


2152 


2233 


2313 




640 


2394 


2474 


2555 


2635 


2715 


2796 


2876 


2956 


3037 


3117 




641 


3197 


3278. 


3358 


3438 


3518 


3598 


3679 


3759 


3839 


3919 




542 


3999 


4079 


4160 


4240 


4320 


4400 


4480 


4560 


4640 


4720 




543 


4800 


4880 


4960 


6040 


5120 


5200 


5279 


5359 


5439 


6519 




544 


6599 


5679 


6759 


5838 


5918 


5998 


6078 


6157 


6237 


6317 




545 


6397 


6476 


6556 


6636 


6715 


6795 


6874 


6954 


7034 


7113 




546 


7193 


7272 


7352 


7431 


7511 


7590 


7670 


7749 


7829 


7908 




547 


7987 


8067 


8146 


8225 


8305 


8384 


8463 


8543 


8622 


8701 




548 


8781 


8860 


8939 


9018 


9097 


9177 


9256 


9335 


9414 


9493 




• 549 


9572 


9651 


9731 


9810 


9889 


9968 


..47 


.126 


.205 


.284 





17 





12 


LOGARITHMS 








N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




550 


740363 


0442 


0521 


0560 


0678 


0757 


0836 


0915 


0994 


1073 




551 


1152 


1230 


1309 


1388 


1467 


1546 


1624 


1703 


1782 


1860 




552 


1939 


2018 


2096 


2175 


2254 


2332 


2411 


2489 


2568 


2646 




553 


2725 


2804 


2882 


2961 


3039 


3118 


3196 


3275 


3353 


3431 




554 


3510 


3558 


3667 


3746 


3823 


3902 


3980 


4058 


4136 


4215 




555 


4293 


4371 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 




556 


5075 


5153 


5231 


5309 


6387 


5465 


5543 


6621 


6699 


5777 




557 


5855 


6933 


6011 


6089 


6167 


6245 


6323 


6401 


6479 


6556 




658 


6634 


6712 


6790 


6868 


6945 


7023 


7101 


7179 


7256 


7334 




559 


7412 


7489 


7567 


7645 


7722 


7800 


7878 


7955 


8033 


8110 




560 


8188 


8266 


8343 


8421 


8498 


8576 


8653 


8731 


8808 


8885 


1 


561 


8963 


9040 


9118 


9195 


9272 


9350 


9427 


9504 


9582 


9659 


[ 


562 


9736 


9814 


9891 


9968 


..45 


.123 


.200 


.277 


.354 


.431 


[ 


563 


750508 


0586 


0663 


0740 


0817 


0894 


0971 


1048 


11-25 


1202 




564 


1279 


1366 


1433 


1510 


1587 


1664 


1741 


1818 


1895 


1972 




565 


2048 


2125 


2202 


2279 


2356 


2433 


2509 


2586 


2663 


£740 




566 


2816 


2893 


2970 


3047 


3123 


3200 


3277 


3353 


3430 


3506 j 




567 


3582 


3660 


3736 


3813 


3889 


3966 


4042 


4119 


4195 


4272 


568 


4348 


4425 


4501 


4578 


4654 


4730 


4807 


4883 


4960 


5036 




569 


6112 


5189 


5265 


5341 


5417 


5494 


5570 


5646 


5722 


6799 




570 


5875 


5951 


6027 


6103 


6180 


6256 


6332 


6408 


6484 


6560 




571 


6636 


6712 


6788 


6864 


6940 


7016 


7092 


7168 


7244 


7320 




672 


7396 


7472 


7548 


7624 


7700 


7775 


7851 


7927 


8003 


8079 I 




573 


8155 


8230 


8306 


8382 


8458 


8533 


8609 


8685 


8761 


6836 "J 




574 


8912 


8988 


9068 


9139 


9214 


9290 


9366 


9441 


9517 


9592 | 




575 


9668 


9743 


9819 


9894 


9970 


..45 


.121 


.196 


.272 


.347 




576 


760422 


0498 


0573 


0649 


0724 


0799 


0875 


0950 


1025 


1101 




577 


1176 


1251 


1326 


1402 


1477 


1552 


1627 


1702 


1778 


1853 




578 


1928 


2003 


2078 


2153 


2228 


2303 


2378 


2453 


2529 


2604 




579 


2679 


2754 


2829 


2904 


2978, 


3053 


3128 


2203 


3278 


3353 




580 


3428 


3503 


3578 


3653 


3727 


3802 


3877 


3952 


4027 


4101 




581 


4176 


4251 


4326 


4400 


4475 


4550 


4624 


4699 


4774 


4848 




582 


4923 


4998 


5072 


5147 


5221 


5296 


5370 


6445 


5520 


5594 




683 


6669 


5743 


5818 


5892 


5966 


6041 


6115 


6190 


6264 


6338 




684 


6413 


6487 


6562 


6636 


6710 


6785 


6869 


6933 


7007 


7082 




585 


7156 


7230 


7304 


7379 


7453 


7527 


7601 


7675 


7749 


7823 




586 


7898 


7972 


8046 


8120 


8194 


8268 


8342 


8416 


8490 


8564 




587 


8638 


8712 


8786 


8860 


8934 


9008 


9082 


9156 


9230 


9303 




588 


9377 


9451 


9525 


9599 


9673 


9746 


9820 


9894 


9968 


..42 




589 


770115 


0189 


0263 


0336 


0410 


0484 


0567 


0631 


0705 


0778 
1514 




590 


0852 


0926 


0999 


1073 


1146 


1220 


1293 


1367 


1440 




691 


1587 


1661 


1734 


1808 


1881 


1955 


2028 


2102 


2175 


2248 




692 


2322 


2395 


2468 


3542 


2615 


2688 


2762 


2835 


2908 


2981 




593 


3055 


3128 


3201 


3274 


3348 


3421 


3494 


3567 


3640 


3713 




694 


3786 


3860 


3933 


4006 


4079 


4152 


4225 


4298 


4371 


4444 




595 


4517 


4590 


4663 


4736 


4809 


4882 


4955 


5028 


5100 


5173 




596 


5246 


6319 


5392 


5465 


5538 


5610 


5683 


5756 


6829 


5902 ■■ 




597 


6974 


6047 


6120 


6193 


6265 


6338 


6411 


6483 


6556 


6629 




598 


6701 


6774 


6846 


6919 


6992 


7064 


7137 


7209 


7282 


7354 




599 


7427 


7499 


7572 


7644 


7717 


7789 


7862 


7934 


8006 


8079 





OF NUMBERS. 


13 


N, 





1 


2 


3 


4 


5 


6 


7 


8 


9 


600 


778151 


8224 


8296 


8368 


8441 


8513 


8585 


8658 


8730 


8802 


601 


8874 


8947 


9019 


9091 


9163 


9236 


9308 


9380 


9452 


9524 


602 


9596 


6669 


9741 


9813 


9885 


9957 


..29 


.101 


.173 


.245 


603 


780317 


0389 


0461 


0533 


0605 


0677 


0749 


0821 


0893 


0905 
1084 f 


604 


1037 


1109 


1181 


1253 


1324 


1396 


1468 


1540 


1612 


605 


1755 


1827 


1899 


1971 


2042 


2114 


2186 


2258 


2329 


2401 


606 


2473 


2544 


2616 


2688 


2759 


2831 


2902 


2974 


3040 


3117 S 


607 


3189 


8260 


3332 


3403 


3475 


3546 


3618 


3689 


3701 


3832 


608 


3904 


3975 


4046 


4118 


4189 


4261 


4332 


4403 


4475 


4646 | 


609 


4617 


4689 


4760 


4831 


4902 


4974 


5045 


5110 


5187 


5259 | 


610 


5330 


5401 


5472 


5543 


5615 


6686 


5757 


5828 


5S99 


5970 


611 


6041 


6112 


6183 


6254 


6325 


6396 


6407 


6538 


6609 


6680 
7890 . 


612 


6751 


6822 


6893 


6964 


7035 


7106 


7177 


7248 


7319 


613 


7460 


7531 


7603 


7673 


7744 


7815 


7885 


7950 


8027 


8098 


614 


8168 


8239 


8310 


8381 


8451 


8522 


8593 


8663 


8734 


8804 


615 


8875 


8946 


9016 


9087 


9157 


9228 


9299 


9369 


9440 


9510 


616 


9581 


9651 


9722 


9792 


9863 


9933 


...4 


. .74 


.144 


.215 


617 


790285 


0356 


0426 


0496 


0507 


0637 


0707 


0778 


08-18 


0918 


618 


0988 


1059 


1129 


1199 


1269 


1340 


1410 


1480 


1550 


1-630 


619 


1691 


1761 


1831 


1901 


1971 


2041 


2111 


2181 


2252 


2322 


620 


2392 


2462 


2532 


2602 


2672 


2742 


2812 


2882 


2952 


8022 


621 


3092 


3162 


3231 


3301 


3371 


3441 


3511 


3581 


3651 


3721 


622 


3790 


3860 


3930 


4000 


4070 


4139 


4209 


4279 


4349 


4418 


623 


4488 


4558 


4627 


4697 


4767 


4836 


4906 


4976 


5045 


5115 


624 


5185 


5254 


5324 


5393 


5463 


5532 


5602 


5672 


5741 


5811 


625 


5880 


5949 


6019 


6088 


6158 


6227 


6297 


6366 


6436 


6505 


626 


6574 


6644 


6713 


6782 


6852 


6921 


6990 


7000 


7129 


7198 


627 


7268 


7337 


7406 


7475 


7545 


7614 


7683 


7752 


7821 


7890 


628 


7960 


8029 


8098 


8167 


8236 


8306 


8374 


8443 


8513 


8582 


629 


8651 


8720 


8789 


8858 


8927 


8996 


9065 


6134 


9203 


9272 


630 


9341 


9409 


9478 


9547 


9610 


9686 


9754 


9823 


9892 


9961 | 


631 


800026 


0098 


0167 


0236 


0305 


0373 


0442 


0511 


0580 


0048 


032 


0717 


0786 


0854 


0923 


0992 


1061 


1129 


1198 


1266 


1335 | 


633 


1404 


1472 


1541 


1609 


1678 


1747 


1815 


1884 


1952 


2021 


634 


2089 


2158 


2226 


2295 


2363 


2432 


2600 


2568 


2637 


2705 


635 


2774 


2842 


2910 


2979 


3047 


3116 


3184 


3252 


3321 


3389 


636 


3457 


3525 


3594 


3662 


3730 


3798 


8867 


3935 


4003 


4071 


637 


4139 


4208 


4276 


4354 


4412 


4480 


4548 


4616 


4685 


4753 


638 


4821 


4889 


4957 


5025 


5093 


6161 


5229 


6297 


6365 


6433 


639 


5501 


5669 


5637 


5705 


6773 


6841 


5908 


6976 


6044 


6112 


640 


6180 


6248 


6316 


6384 


6451 


6519 


6587 


6655 


6723 


6790 | 


641 


6858 


6926 


6994 


7061 


7129 


7167 


7264 


7332 


7400 


7467 \ 
8143 { 


642 


7535 


7603 


7670 


7738 


7806 


7873 


7941 


8008 


8076 


643 


8211 


8279 


8346 


8414 


8481 


8549 


8616 


8684 


8751 


8818 \ 


64-1 


8886 


8953 


9021 


9088 


9156 


9223 


9290 


9358 


9425 


9492 j 


645 


9500 


9627 


9694 


9762 


9829 


9896 


9964 


..31 


..98 


.165 j 
0337 


646 


810233 


0300 


0367 


0434 


0501 


0596 


0636 


0703 


0770 


647 


0904 


(1971 


1039 


1106 


1173 


1240 


1307 


1374 


1441 


1508 


648 


1675 


1642 


1709 


1776 


1843 


1910 


1977 


2044 


2111 


2178 


649 


2245 


2312 


2379 


2445 


2512 


2579 


2646 


2713 


2780 


2847 



14 


LOGARITHMS 


N. 





1 


o 


3 


4 


5 


6 


7 


8 


9 


650 


812913 


2980 


3047 


3114 


3181 


3247 


3314 


3381 


3448 


3514 


651 


3581 


3648 


3714 


3781 


3848 


3914 


3981 


4048 


4114 


4181 


652 


4248 


4314 


4381 


4447 


4514 


4581 


4647 


4714 


4780 


4847 


653 


4913 


4980 


5046 


5113 


5179 


5246 


5312 


5378 


5445 


6511 


654 


5578 


5644 


5711 


5777 


5843 


6910 


5976 


6042 


6109 


6175 


655 


6241 


6308 


6374 


6440 


6506 


6573 


6639 


6705 


6771 


6S38 


656 


6904 


6970 


7036 


7102 


7169 


7233 


7301 


7367 


7433 


7499 


657 


7565 


7631 


7698 


7764 


7830 


7896 


7962 


8028 


8094 


8160 


658 


8226 


8292 


8358 


8424 


8490 


8556 


8622 


8688 


8754 


8820 


659 


8885 


8951 


9017 


9083 


9149 


9215 


9281 


9346 


9412 


9478 


660 


9544 


9610 


9676 


9741 


9807 


9873 


9939 


..A 


..70 


.130 


661 


820201 


0267 


0333 


0399 


0464 


0530 


0595 


0661 


0727 


0792 


662 


0858 


0924 


0989 


1055 


1120 


1186 


1251 


1317 


1382 


1448 


663 


1514 


1579 


1645 


1710 


1776 


1841 


1906 


1972 


2037 


2103 


664 


2168 


2233 


2299 


2364 


2430 


2495 


2560 


2626 


2691 


2756 


665 


2822 


2887 


2952 


3018 


3083 


3148 


3213 


3279 


3344 


3409 


666 


3474 


3539 


3605 


3670 


3735 


3800 


3865 


3930 


3996 


4061 


667 


4126 


4191 


4256 


4321 


4386 


4451 


4516 


4681 


4646 


4711 


668 


4776 


4841 


4906 


4971 


5036 


5101 


5166 


5231 


5296 


5361 


669 


5426 


5491 


5556 


5621 


5686 


5751 


5816 


6880 


5945 


6010 


670 


6075 


6140 


6204 


6269 


6334 


6399 


6464 


6528 


6593 


6658 


671 


6723 


6787 


6852 


6917 


6981 


7046 


7111 


7175 


7240 


7305 


672 


7369 


7434 


7499 


7563 


7628 


7692 


7757 


7821 


7886 


7951 


673 


8015 


8080 


8144 


8209 


8273 


8338 


8402 


8467 


8531 


8595 


674 


8660 


8724 


8789 


8853 


8918 


8982 


9046 


9111 


9175 


9239 


675 


9304 


9368 


9432 


9497 


9561 


9625 


9690 


9754 


9818 


9882 


676 


9947 


..11 


..75 


.139 


.204 


.268 


.332 


.396 


.460 


.626 


677 


830589 


0653 


0717 


0781 


0845 


0909 


0973 


1037 


1102 


1166 


678 


1230 


1294 


1358 


1422 


1486 


1550 


1614 


1678 


1742 


1806 


679 


1870 


1934 


1998 


2062 


2126 


2189 


2253 


2317 


2381 


2445 


680 


2509 


2573 


2637 


2700 


2764 


2828 


2892 


2956 


3020 


3083 


681 


3147 


3211 


3275 


3338 


3402 


3466 


3530 


3593 


3657 


3721 


682 


3784 


3848 


3912 


3975 


4039 


4103 


4166 


4230 


4294 


4357 


683 


4421 


4484 


4548 


4611 


4675 


4739 


4802 


4866 


4929 


4993 


684 


5056 


5120 


5183 


5247 


5310 


6373 


5437 


6500 


6564 


6627 


685 


5691 


5754 


5817 


5881 


5944 


6007 


6071 


6134 


6197 


6261 


686 


6324 


6387 


6451 


6514 


6577 


6641 


6704 


6767 


6830 


6894 


687 


6957 


7020 


7083 


7146 


7210 


7273 


7336 


7399 


7462 


7525 


688 


7588 


7652 


7715 


7778 


7841 


7904 


7967 


8030 


8093 


8156 


689 


8219 


8282 


8345 


8408 


8471 


8534 


8597 


8660 


8723 


8786 
9415 


690 


8849 


8912 


8975 


9038 


9109 


9164 


9227 


9289 


9352 


691 


9478 


9541 


9604 


9667 


9729 


9792 


9855 


9918 


9981 


..43 


692 


840106 


0169 


0232 


0294 


0357 


0420 


0482 


0545 


0608 


0671 


693 


0733 


0796 


0859 


0921 


0984 


1046 


1109 


1172 


1234 


129 1 


694 


1359 


1422 


1485 


1547 


1610 


1672 


1735 


1797 


1860 


1922 


695 


1985 


2047 


2110 


2172 


2235 


2297 


2360 


2422 


2484 


2547 


696 


2609 


2672 


2734 


2796 


2859 


2921 


2983 


3046 


3108 


3170 


697 


3233 


3295 


3357 


3420 


3482 


3544 


3606 


3669 


3731 


3793 


698 


3855 


3918 3980 


4042 


4104 


4166 


4229 


4291 


4353 


4415 


699 


4477 


4539 4601 


4664 


4726 


4788 


4850 


4912 


4:r. ; 4 


5036 



OF NUMBERS. 15 


N. 


| , 


2 


3 


4 


5 


6 


7 


8 


9 


700 


845098 


5160 


5222 


5284 


5346 


5408 


5470 


5532 


6594 


5656 


701 


5718 


5780 


5842 


5904 


5966 


6028 


6090 


6151 


6213 


6275 


702 


6337 


6399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6894 


703 


6955 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 


7511 


704 


7573 


7634 


7676 


7758 


7819 


7831 


7943 


8004 


8066 


8128 


705 


8189 


8251 


8312 


8374 


8435 


8497 


8559 


8620 


8682 


8743 


706 


8805 


8866 


8928 


8989 


9051 


9112 


9174 


9235 


9297 


9358 


| 707 


9419 


9481 


9542 


9604 


9665 


9726 


9788 


9849 


9911 


9972 


703 


850033 


0095 


0156 


0217 


0279 


0340 


0401 


0462 


0524 


0585 


709 


0646 


0707 


0769 


0830 


0891 


0952 


1014 


1075 


1136 


1197 


710 


1258 


1320 


1381 


1442 


1503 


1564 


1625 


1686 


1747 


1809 


1 711 


1870 


1931 


1992 


2053 


2114 


2175 


2236 


2297 


2358 


2419 


712 


2480 


2541 


2602 


2663 


2724 


2785 


2846 


2907 


2968 


3029 


713 


3090 


3150 


3211 


3272 


3333 


3394 


3455 


3516 


3577 


3637 


714 


3698 


3759 


3820 


3881 


3941 


4002 


4063 


4124 


4185 


4245 


715 


4306 


4367 


4428 


4488 


4549 


4610 


4670 


4731 


4792 


4852 


716 


4913 


4974 


5034 


5095 


5156 


5216 


5277 


5337 


5398 


5459 


717 


5519 


5580 


5640 


5701 


5761 


5822 


5882 


5943 


6003 


6064 


718 


6124 


6185 


6245 


6306 


6366 


6427 


6487 


6548 


6608 


6668 


719 


6729 


6789 


6850 


6910 


6970 


7031 


7091 


7152 


7212 


7272 


720 


7332 


7393 


7453 


7513 


7574 


7634 


7694 


7755 


7815 


7875 


721 


7935 


7995 


8056 


8116 


8176 


8236 


8297 


8357 


8417 


8477 


722 


8537 


8597 


8657 


8718 


8778 


8838 


8898 


8958 


9018 


9078 


723 


9138 


9198 


9258 


9318 


9379 


9439 


9499 


9559 


9619 


'9679 


724 


9739 


9799 


9859 


9918 


9978 


..38 


..98 


.158 


.218 


.278 


725 


860338 


0398 


0458 


0518 


0578 


0637 


0697 


0757 


0817 


0877 


726 


0937 


0996 


1056 


1116 


1176 


1236 


1295 


1355 


1415 


1475 


727 


1534 


1594 


1664 


1714 


1773 


1833 


1893 


1952 


2012 


2072 


728 


2131 


2191 


2261 


2310 


2370 


2430 


2489 


2549 


2608 


2668 


729 


2728 


2787 


2847 


2906 


2966 


3025 


3085 


3144 


3204 


3263 


730 


3323 


3382 


3442 


3501 


3561 


3620 


3680 


3739 


3799 


3858 


731 


3917 


3977 


4036 


4096 


4165 


4214 


4274 


4333 


4392 


4452 


732 


4511 


4570 


4630 


4689 


4148 


4808 


4867 


4926 


4985 


5045 


733 


5104 


5163 


5222 


5282 


5341 


5400 


5459 


5519 


5578 


5637 


734 


5696 


5755 


5814 


5874 


5933 


5992 


6051 


6110 


6169 


6228 


735 


6287 


6346 


6405 


6465 


6524 


6583 


6642 


6701 


6760 


6819 


736 


6878 


6937 


6996 


7055 


7114 


7173 


7232 


7291 


7350 


7409 


737 


7467 


7526 


7685 


7644 


7703 


7762 


7821 


7880 


7939 


7998 


738 


8056 


8115 


8174 


8233 


8292 


8350 


8409 


8468 


8527 


8586 


739 


8644 


8703 


8762 


8821 


8879 


8938 


8997 


9056 


9114 


9173 


740 


9232 


9290 


9349 


9408 


9466 


9625 


9584 


9642 


9701 


9760 | 


741 


9818 


9877 


9935 


9994 


..63 


.111 


.170 


.228 


.287 


.345 j^ 


742 


870404 


0462 


0521 


0679 


0638 


0696 


0755 


0813 


0872 


0930 j 


743 


0989 


1047 


1106 


1164 


1223 


1281 


1339 


1398 


1456 


1515 | 


744 


1573 


1631 


1690 


1748 


1806 


1865 


1923 


1981 


2040 


2098 


745 


2156 


2215 


2273 


2331 


2389 


2448 


2506 


2564 


2622 


2681 


746 


2739 


2797 


2855 


2913 


2972 


3030 


3088 


3146 


3204 


3262 


747 


3321 


3379 


3437 


3495 


3553 


3611 


3669 


3727 


3785 


3844 


748 


3902 


3960 


4018 


4070 


4134 


4192 


4250 


4308 


4360 


4424 


749 


4482 


4540 


4598 


4656 


4714 


4772 


4830 


4888 


4945 


5003 



16 




LOGARITHMS 


N. 





1 


2 


3 j 4 


5 


6 


7 


8 


9 


750 


875061 


5119 


6177 


5235 


5293 


5351 


5409 


5466 


5524 


5582 


751 


5640 


6698 


5756 


5813 


5871 


5929 


5987 


6045 


6102 


6160 


752 


6218 


6276 


6333 


6391 


6449 


6507 


6564 


6622 


6680 


6737 


753 


6795 


6853 


6910 


'6968 


7026 


7083 


7141 


7199 


7256 


7314 


754 


7371 


7429 


7487 


7544 


7602 


7659 


7717 


7774 


7832 


7889 


755 


7947 


8004 


8062 


8119 


8177 


8234 


8292 


8349 


8407 


8464 


756 


8522 


8579 


8637 


8694 


8752 


8809 


8866 


8924 


8981 


9039 


757 


9096 


9153 


9211 


9268 


9325 


9383 


9440 


9497 


9555 


9612 


758 


9669 


9726 


9784 


9841 


9898 


9956 


..13 


..70 


.127 


.185 


759 


880242 


0299 


0356 


0413 


0471 


0528 


0580 


0642 


0699 


0756 


760 


0814 


0871 


0928 


0985 


1042 


1099 


1156 


1213 


1271 


1328 


761 


1385 


1442 


1499 


1556 


1613 


1670 


1727 


1784 


1841 


1898 


762 


1955 


2012 


2069 


2126 


2183 


2240 


2297 


2354 


2411 


2468 


763 


2525 


2581 


2638 


2695 


2752 


2809 


2866 


2923 


2980 


3037 


764 


3093 


3150 


3207 


3264 


3321 


3377 


3434 


3491 


3548 


3605 


765 


3661 


3718 


3775 


3832 


3888 


3945 


4002 


4059 


4115 


4172 


766 


4229 


4285 


4342 


4399 


4455 


4512 


4569 


4625 


4682 


4739 


767 


4795 


4852 


4909 


4966 


5022 


5078 


5135 


5192 


5248 


5305 


768 


5361 


5418 


5474 


5531 


5587 


5644 


5700 


5757 


5813 


5870 


769 


5926 


6983 


6039 


6096 


6152 


6209 


6265 


6321 


6378 


6434 


770 


6491 


6547 


6604 


6660 


6716 


6773 


6829 


6885 


6942 


6998 


771 


7054 


7111 


7167 


7233 


7280 


7336 


7392 


7449 


7506 


7561 


772 


7617 


7674 


7730 


7786 


7842 


7898 


7955 


8011 


8067 


8123 


773 


8179 


8236 


8292 


8348 


8404 


8460 


8516 


8573 


8629 


8655 


774 


8741 


8797 


8853 


8909 


8965 


9021 


9077 


9134 


9190 


9246 


775 


9302 


9358 


9414 


9470 


9526 


9582 


9638 


9694 


9750 


9806 


776 


9862 


9918 


0974 


..30 


..86 


.141 


.197 


.253 


.309 


.365 


777 


890421 


0477 


0533 


0589 


0645 


0700 


0756 


0812 


0868 


0924 


778 


0980 


1035 


1091 


1147 


1203 


1259 


1314 


1370 


1426 


1482 ! 


779 


1537 


1593 


1649 


1705 


1760 


1816 


1872 


1928 


1983 


2039 


780 


2095 


2150 


2206 


2262 


2317 


2373 


2429 


2484 


2540 


2595 


781 


2651 


2707 


2762 


2818 


2873 


2929 


2985 


3040 


3096 


3151 


782 


3207 


3262 


3318 


3373 


3429 


3484 


3540 


3595 


3651 


3706 


783 


3762 


3817 


3873 


3928 


3984 


4039 


4094 


4150 


4205 


4261 


784 


4316 


4371 


4427 


4482 


4538 


4593 


4648 


4704 


4759 


4814 


785 


4870 


4925 


4980 


5036 


5091 


5146 


5201 


5257 


5312 


5367 


786 


5423 


5478 


5533 


5588 


5644 


5699 


5754 


5809 


5884 


5920 


787 


5975 


6030 


6085 


6140 


6195 


6251 


6306 


6361 


6416 


6471 


788 


6526 


6581 


6636 


6692 


6747 


6802 


6857 


6912 


6967 


7 022 


789 


7077 


7132 


7187 


7242 


7297 


7352 


7407 


7462 


7517 


7 o72 


790 


7627 


7683 


7737 


7792 


7847 


7902 


7957 


8012 


8067 


8122 


791 
792 


8176 


8231 


8286 


8341 


8396 


8451 


8506 


8561 


8615 


8670 


8725 


8780 


8835 


8890 


8944 


8999 


9054 


9109 


9164 


9218 


793 


9273 


9328 


9383 


9437 


9492 


9547 


9602 


9656 


9711 


9766 


794 


9821 


9875 


9930 


9985 


..39 


..94 


.149 


.203 


.258 


.312 


795 


900367 


0422 


0476 


0531 


0586 


0640 


0695 


0749 


0804 


0859 


796 


0913 


0968 


1022 


1077 


1131 


1186 


1240 


1295 


1349 1 1-104 


797 


1458 


1513 


1567 


1622 


1676 


1736 


1785 


1840 . 


l^.v4 1948 


798 


2003 


2057 


2112 


2166 


2221 


2275 


2329 


2384 2438 ! 2492 


799 


2547 


2601 


2655 


2710 


2764 


2818 


2b 7 3 


2927 2"81 '6iM 













=1 






F NUMBERS 






17 1! 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


o 


800 


903090 


3144 


3199 


3253 


3307 


3361 


3416 


3470 


3524 


3578 


801 


3633 


3687 


3741 


3795 


3849 


3904 


3958 


4012 


4066 


4120 


802 


4174 


4229 


4283 


4337 


4391 


4445 


4499 


4553 


4807 


4661 


803 


4716 


4770 


4824 


4878 


4932 


4986 


5040 


5094 


5148 


5202 


804 


5256 


5310 


5364 


5418 


5472 


5526 


5580 


5634 


6688 


5742 


ao5 


5796 


5850 


5904 


5958 


6012 


6066 


6119 


6173 


6227 


i 
6281 


803 


6335 


6389 


6443 


6497 


6551 


6604 


6658 


6712 


6766 


6820 


807 


6874 


6927 


6981 


7035 


7089 


7143 


7196 


7250 


7304 


7358 


808 


7411 


7465 


7519 


7573 


7626 


7680 


7734 


7787 


7841 


7895 


809 


7949 


8002 


8056 


8110 


8163 


8217 


8270 


8324 


8378 


8431 


810 


8485 


8539 


8592 


8646 


8699 


8753 


8807 


8860 


8914 


8967 


811 


9021 


9074 


9128 


9181 


9235 


9289 


9342 


9396 


9449 


9503 


812 


9556 


9610 


9663 


9716 


9770 


9823 


9877 


9930 


9984 


..37 : 


813 


910091 


0144 


0197 


0251 


0304 


0358 


0411 


0464 


0518 


0571 


814 


0824 


0878 


0731 


0784 


0838 


0891 


0944 


0998 


1051 


1104 | 


i 815 


1158 


1211 


1264 


1317 


1371 


1424 


1477 


1530 


1584 


1637 i 


816 


1690 


1743 


1797 


1850 


1903 


1956 


2009 


2063 


2115 


2169 ! 


817 


2222 


2275 


2323 


2381 


2435 


2488 


2541 


2594 


2645 


2700 ! 


818 


2753 


2806 


2859 


2913 


2966 


3019 


3072 


3125 


3178 


3231 


819 


3284 


3337 


3390 


3443 


3496 


3649 


3602 


3655 


3708 


3761 


820 


3814 


3867 


3920 


3973 


4026 


4079 


4132 


4184 


4237 


4290 


821 


4343 


4396 


4449 


4502 


4555 


4608 


4660 


4713 


4766 


4819 


822 


4872 


4925 


4977 


5030 


6083 


5136 


5189 


5241 


5594 


5347 


823 


6400 


5453 


5505 


5558 


5611 


5664 


5716 


5769 


5822 


5875 | 


! 824 


5927 


5980 


6033 


6085 


6138 


6191 


6243 


6296 


6349 


6401 | 
6927 


825 


6454 


6507 


6559 


6612 


6664 


6717 


6770 


6822 


6875 


826 


6980 


7033 


7085 


7138 


7190 


7243 


7295 


7348 


7400 


7453 


827 


7506 


7558 


7611 


7663 


7716 


7768 


7820 


7873 


7925 


7978 


828 


8030 


8033 


8185 


8188 


8240 


8293 


8345 


8397 


8450 


8502 


829 


8555 


8607 


8659 


8712 


8764 


8816 


8869 


8921 


8973 


9026 


830 
| 831 


9078 


9130 


9183 


9235 


9287 


9340 


9392 


9444 


9496 


9549 I 


9601 


9653 


9706 


9758 


9810 


9862 


9914 


9967 


..19 


..71 ! 


| 832 


920123 


0176 


0228 


0280 


0332 


0384 


0436 


0489 


0541 


0593 ! 


j 833 


0845 


0697 


0749 


0801 


0853 


0903 


0958 


1010 


1062 


1114 


\ 834 


1166 


1218 


1270 


1322 


1374 


1426 


1478 


1530 


1582 


1634 


! 835 


1686 


1738 


1790 


1842 


1894 


1946 


1998 


2050 


2102 


2154 


j 836 


2206 


2258 


2310 


2362 


2414 


2466 


2518 


2570 


2622 


26 74 \ 


837 


2725 


2777 


2829 


2881 


2933 


2985 


3037 


3089 


3140 


3192 


838 


3244 


3296 


3348 


3399 


3451 


3503 


3555 


3607 


3658 


3710 \ 


839 


3762 


3814 


3865 


3917 


3969 


4021 


4072 


4124 


4147 


4228 j 

4744 
6201 \\ 


840 


4279 


4331 


4383 


4434 


4486 


4538 


4589. 


4641 


4693 


841 


4796 


4848 


4899 


4951 


5003 


5054 


5108 


6157 


5209 


842 


5312 


5364 


5415 


5467 


5518 


5570 


5621 


5673 


6725 


57 76 |j 


843 


5828 


5874 


5931 


6982 


6034 


6085 


6137 


6188 


6240 


6291 ll 


844 


6342 


6394 


6445 


6497 


6548 


6600 


6651 


6702 


6754 


6805 


845 


6857 


6908 


6959 


7011 


7062 


7114 


7165 


7216 


7268 


7319 ! 


846 


7370 


7422 


7473 


7624 


7576 


7627 


7678 


7730 


7783 


7832 


847 


7883 


7935 


7986 


8037 


8088 


8140 


8191 


8242 


8293 


8345 


848 


8396 


8447 


8498 


8549 


8601 


8652 


8703 


8754 


880j 


8S57 ; 


849 


8903 


8959 


9010 


9081 


9112 


9163 


9216 


9266 


9317 


93ti8 | 

j 



18 




LOGARITHMS 




N. 





I 


2 


3 


4 


5 


6 


7 


8 


9 




850 


929419 


9473 


9521. 


9572 


9623 


9674 


9725 


9776 


9827 


9879 




851 


9930 


9981 


..32 


..83 


.134 


.185 


.236 


.287 


.338 


.389 




852 


930440 


0491 


0542 


0592 


0643 


0694 


0745 


0796 


0847 


0898 




853 


0949 


1000 


1051 


1102 


1153 


1204 


1254 


1305 


1356 


1407 




854 


1458 


1509 


1560 


1610 


1661 


1712 


1763 


1814 


1865 


1915 




855 


1966 


2017 


2068 


2118 


2169 


2220 


2271 


2322 


2372 


2423 




856 


2474 


2524 


2575 


2626 


2677 


2727 


2778 


2829 


2879 


2930 




857 


2981 


3031 


3082 


3133 


3183 


3234 


3285 


3335 


3386 


3437 




858 


3487 


3538 


3589 


3639 


3690 


3740 


3791 


3841 


3892 


3943 




859 


3993 


4044 


4094 


4145 


4195 


4246 


4269 


4347 


4397 


4448 




860 


4498 


4549 


4599 


4650 


4700 


4751 


4801 


4852 


4902 


4953 




861 


5003 


5054 


5104 


5154 


5205 


5255 


5306 


5356 


5406 


5457 




862 


5507 


5558 


5608 


5658 


5709 


5769 


5809 


5860 


5910 


5960 




863 


6011 


6061 


6111 


6162 


6212 


6262 


6313 


6363 


6413 


6463 




864 


6514 


6564 


6614 


6665 


6715 


6765 


6816 


6865 


6916 


6966 




865 


7016 


7066 


7117 


7167 


7217 


7267 


7317 


7367 


7418 


7468 




866 


7518 


7568 


7618 


7668 


7718 


7769 


7819 


7869 


7919 


7969 




867 


8019 


8069 


8119 


8169 


8219 


8269 


8320 


8370 


8420 


8470 




868 


8520 


8570 


8620 


8670 


8720 


8770 


8820 


8870 


8919 


8970 




869 


9020 


9070 


9120 


9170 


9220 


9270 


9320 


9369 


9419 


9469 




870 


9519 


9569 


9616 


9669 


9719 


9769 


9819 


9869 


9918 


9968 




871 


940018 


0068 


0118 


0168 


0218 


0267 


0317 


0367 


0417 


0467 




872 


0516 


0566 


0616 


0666 


0716 


0765 


0815 


0865 


0915 


0964 




873 


1014 


1064 


1114 


1163 


1213 


1263 


1313 


1362 


1412 


1462 




874 


1511 


1561 


1611 


1660 


1710 


1760 


1809 


1859 


1909 


1958 




875 


2008 


2058 


2107 


2157 


2207 


2256 


2306 


2355 


2405 


2455 




876 


2504 


2554 


2603 


2653 


2702 


2752 


2801 


2851 


2901 


2950 




877 


3000 


3049 


3099 


3148 


3198 


3247 


3297 


3346 


3396 


3445 




878 


3495 


3544 


3593 


3643 


3692 


3742 


3791 


3841 


3890 


3939 




879 


3989 


4038 


4088 


4137 


4186 


4236 


4285 


4335 


4384 


4433 




880 


4483 


4532 


4581 


4631 


4680 


4729 


4779 


4828 


4877 


4927 




881 


4976 


5025 


5074 


5124 


5173 


5222 


5272 


5321 


5370 


5419 




882 


5469 


5518 


5567 


5616 


5665 


5715 


5764 


5813 


5862 


5912 




883 


5961 


6010 


6059 


6108 


6157 


6207 


6256 


6305 


6354 


6403 




884 


6452 


6501 


6551 


6600 


6649 


6698 


6747 


6796 


6845 


6894 




885 


6943 


6992 


7041 


7090 


7140 


7189 


7238 


7287 


7336 


7385 




886 


7434 


7483 


7532 


7581 


7630 


7679 


7728 


7777 


7826 


7875 




887 


7924 


7973 


8022 


8070 


8119 


8168 


8217 


8266 


8315 


8365 




888 


8413 


8462 


8611 


8560 


8609 


8657 


8706 


8755 


88U4 


8853 




889 


8902 


8951 


8999 


9048 


9097 


9146 


9195 


9244 


9292 


9341 




890 


9390 


9439 


9488 


9536 


9585 


9634 


9683 


9731 


9780 


9829 




891 


9878 


9926 


9975 


..24 


..73 


.121 


.170 


.219 


.267 


.316 




892 


950365 


0414 


0462 


0511 


0560 


0608 


0657 


0706 


0754 


0803 




893 


0851 


0900 


0949 


0997 


1046 


1095 


1143 


1192 


1240 


1289 




894 


1338 


1386 


1435 


1483 


1532 


1580 


1629 


1677 


1726 


17/5 




895 


1823 


1872 


1920 


1969 


2017 


2066 


2114 


2163 


2211 


2260 




896 


2308 


2356 


2405 


2453 


2502 


2550 


2599 


2647 


5696 


2744 




897 


2792 


2841 


2889 


2938 


2986 


3034 


3083 


3131 


3180 


3228 




898 


3276 


3325 


3373 


3421 


3470 


3518 


3566 


3615 


3663 


3711 




899 


3760 


3»08 


3856 


3905 


3953 


4001 


4019 


4098 


4146 


4194 











OF NUMBERS. 19 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




900 


954243 


4291 


4339 


4387 


4435 


4484 


4532 


4580 


4628 


4677 




901 


4725 


4773 


4821 


4869 


4918 


4966 


5014 


5062 


5110 


5158 




902 


5207 


5255 


5303 


6351 


5399 


5447 


5495 


5543 


5592 


5640 




903 


5688 


5736 


5784 


5832 


6880 


5928 


5976 


6024 


6072 


6120 




904 


6168 


6216 


6265 


6313 


6361 


6409 


6457 


6505 


6553 


6601 




905 


6649 


6697 


6745 


6793 


6840 


6888 


6936 


6984 


7032 


7080 


906 


7128 


7176 


7224 


7272 


7320 


7368 


7416 


7464 


7512 


7559 




907 


7607 


7655 


7703 


7751 


7799 


7847 


7894 


7942 


7990 


8038 




908 


8086 


8134 


8181 


8229 


8277 


8325 


8373 


8421 


8468 


8516 




909 


8564 


8612 


8659 


8707 


8755 


8803 


8850 


8898 


8946 


8994 




910 


9041 


9089 


9137 


9185 


9232 


9280 


9328 


9375 


9423 


9471 




911 


9518 


9566 


9614 


9661 


9709 


9757 


9804 


9852 


9900 


9947 




912 


9995 


..42 


..90 


.138 


.185 


.233 


.280 


.328 


.376 


.423 




913 


960471 


0518 


0566 


0613 


0661 


0709 


0756 


0804 


0851 


0899 




914 


0946 


0994 


1041 


1089 


1136 


1184 


1231 


1279 


1326 


1374 




915 


1421 


1469 


1516 


1563 


1611 


1658 


1706 


1753 


1801 


1848 




916 


1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2275 


2322 




917 


2369 


2417 


2464 


2511 


2559 


2606 


2653 


2701 


2748 


2795 




918 


2843 


2890 


2937 


2985 


3032 


3079 


3126 


3174 


3221 


3268 




919 


3316 


3363 


3410 


3457 


3504 


3552 


3599 


3646 


3693 


3741 




920 


3788 


3835 


3882 


3929 


3977 


4024 


4071 


4118 


4165 


4212 




921 


4260 


4307 


4354 


4401 


4448 


4495 


4542 


4590 


4637 


4684 




922 


4731 


4778 


4825 


4872 


4919 


4966 


5013 


6061 


5108 


5155 




923 


5202 


5249 


5296 


5343 


5390 


5437 


5484 


5531 


5578 


5625 




924 


5672 


5719 


5766 


5813 


5860 


5907 


5954 


6001 


6048 


6095 




925 


6142 


6189 


6236 


6283 


6329 


6376 


6423 


6470 


6517 


6564 




926 


6611 


6658 


6705 


6752 


6799 


6845 


6892 


6939 


6986 


7033 




927 


7080 


7127 


7173 


7220 


7267 


7314 


7361 


7408 


•7454 


7501 




928 


7548 


7595 


7642 


7688 


7735 


7782 


7829 


7876 


7922 


7969 




929 


8016 


8062 


8109 


8156 


8203 


8249 


8296 


8343 


8390 


8436 




930 


8483 


8530 


8576 


8623 


8670 


8716 


8763 


8810 


8856 


8903 




931 


8950 


8996 


9043 


9090 


9136 


9183 


9229 


9276 


9323 


9369 




932 


9416 


9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9835 




933 


9882 


9928 


9975 


..21 


..68 


.114 


.161 


.207 


.254 


.300 




934 


970347 


0393 


0440 


0486 


0533 


0579 


0626 


0672 


0719 


0765 




935 


0812 


0858 


0904 


0951 


0997 


1044 


1090 


1137 


1183 


1229 




936 


1276 


1322 


1369 


1415 


1461 


1508 


1554 


1601 


1647 


1693 




937 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


2064 


2110 


2157 




938 


2203 


2249 


2295 


2342 


2388 


2434 


2481 


2527 


2573 


2619 




939 


2666 


2712 


2758 


2804 


2851 


2897 


2943 


2989 


3035 


3082 




940 


3128 


3174 


3220 


3266 


3313 


3359 


3405 


3451 


3497 


3543 




941 


3590 


3636 


3682 


3728 


3774 


3820 


3866 


3913 


3959 


4005 




942 


4051 


4097 


4143 


4189 


4235 


4281 


4327 


4374 


4420 


4466 




943 


4512 


4558 


4604 


4650 


4696 


4742 


4788 


4834 


4880 


4926 




944 


4972 


5018 


5064 


6110 


5156 


5202 


5248 


5294 


5340 


5386 




945 


5432 


5478 


5524 


5570 


5616 


5662 


5707 


5753 


5799 


5845 




946 


5891 


5937 


5983 


6029 


6075 


6121 


6167 


6212 


6258 


6304 




947 


6350 


6396 


6442 


6488 


6533 


6579 


6925 


6671 


6717 


6763 




948 


6808 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 




949 


7266 


7312 


7358 


7403 


7449 


7495 


7541 


7586 


7632 


7678 





20 


LOGARITHMS 


1 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


950 


977724 


7769 


7815 


7861 


7906 


7952 


7998 


8043 


8089 


8135 


951 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


8546 


8591 


952 


8637 


8683 


8728 


8774 


8819 


8865 


8911 


8956 


9002 


9047 


953 


9093 


9138 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


9503 


954 


9548 


9594 


9639 


9685 


9730 


9776 


9821 


9867 


9912 


9958 


955 


980003 


0049 


0094 


0140 


0185 


0231 


0276 


0322 


0367 


0412 


956 


0458 


0503 


0549 


0594 


0840 


0885 


0730 


0776 


0821 


0867 


957 


0912 


0957 


1003 


1048 


1093 


1139 


1184 


1229 


1275 


1320 


958 


1366 


1411 


1456 


1501 


1547 


1592 


1637 


1683 


1728 


1773 


959 


1819 


1864 


1909 


1954 


2000 


2045 


2090 


2135 


2181 


2226 


930 


2271 


2316 


2362 


2407 


2452 


2497 


2543 


2588 


2633 


2678 


961 


2723 


2769 


2814 


2859 


2904 


2949 


2994 


3040 


3085 


3130 


962 


3175 


3220 


3265 


3310 


3356 


3401 


3446 


3491 


3536 


3581 


963 


3626 


3671 


3716 


3762 


3807 


3852 


3897 


3942 


3987 


4032 


964 


4077 


4122 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 


965 


4527 


4572 


4617 


4662 


4707 


4752 


4797 


4842 


4887 


4932 


966 


4977 


5022 


5087 


5112 


5157 


5202 


5247 


5292 


5337 


5382 


967 


5426 


5471 


5516 


5561 


5606 


5651 


5699 


5741 


5786 


5830 


968 


5875 


5920 


5965 


6010 


6055 


6100 


6144 


6189 


6234 


6279 


969 


6324 


6369 


6413 


6458 


6503 


6548 


6593 


6637 


6682 


6727 


970 


6772 


6817 


6861 


6908 


6951 


6996 


7040 


7085 


7130 


7175 


971 


7219 


7264 


7309 


7353 


7398 


7443 


7488 


7532 


7577 


7622 


972 


7666 


7711 


7756 


7800 


7845 


7890 


7934 


7979 


8024 


8088 


973 


8113 


8157 


8202 


8247 


8291 


8336 


8381 


8425 


8470 


8514 


974 


8559 


8604 


8648 


8693 


8737 


8782 


8826 


8871 


8916 


8960 


975 


9005 


9049 


9093 


9138 


9183 


9227 


9272 


9316 


9361 


9405 


976 


9450 


9494 


9539 


9583 


9628 


9672 


9717 


9761 


9808 


9850 


977 


9895 


9939 


9983 


..28 


..72 


.117 


.161 


.206 


.250 


.294 


978 


990339 


0383 


0428 


0472 


0516 


0561 


0605 


0650 


0694 


0738 


979 


0783 


0827 


0871 


0916 


0960 


1004 


1049 


1093 


1137 


1182 


980 


1226 


1270 


1315 


1359 


1403 


1448 


1492 


1536 


1580 


1625 


981 


1669 


1713 


1758 


1802 


1846 


1890 


1935 


1979 


2023 


2067 


982 


2111 


2156 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


2509 


983 


2554 


2598 


2642 


2686 


2730 


2774 


2819 


2863 


2907 


2951 


984 


2995 


3039 


3083 


3127 


3172 


3216 


3260 


3304 


3348 


3392 


985 


3436 


3480 


3524 


3568 


3613 


3657 


3701 


3745 


3789 


3833 


986 


3877 


3921 


3965 


4009 


4053 


4097 


4141 


4185 


4229 


4273 


987 


4317 


4361 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


4713 


988 


4757 


4801 


4845 


4886 


4933 


4977 


5021 


5065 


5108 


5152 


989 


5196 


5240 


5284 


5328 


5372 


5416 


5460 


5504 


5547 


5591 


990 


5635 


5679 


5723 


5767 


5811 


5854 


5898 


5942 


5986 


6030 


991 


6074 


6117 


6161 


6205 


6249 


6293 


6337 


6380 


6424 


6468 


992 


6512 


6555 


6599 


6643 


6687 


6731 


6774 


6818 


6862 


6906 


993 


6949 


6993 


7037 


7080 


7124 


7168 


7212 


7255 


7299 


7343 


994 


7386 


7430 


7474 


7517 


7561 


7605 


7648 


7692 


7736 


7779 


995 


7823 


7867 


7910 


7954 


7998 


8041 


8085 


8129 


8172 


8216 


996 


8259 


8303 


8347 


8390 


8434 


8477 


8521 


8564 


8608 


8652 


997 


8695 


8739 


8792 


8826 


8869 


8913 


8956 


9000 


9043 


9087 


998 


9131 


9174 


9218 


9261 


9305 


9348 


9392 


9435 


9479 


9522 


999 

Ik 


9565 


9609 1 9652 


9696 


9739 


9783 


9826 


9870 


9913 


9957 



1 

TABLE II. Log. Sines and Tangents. (0°) Natural Sines. 21 




' 


time. 


D.10" 


Cosine. 


D.10" 


Tang. 


D.10" 


Coiang. 


N.sine 


N. cos. 









Minus inf. 




10.000000 


Minns inf. 




Infinite. 


oooou 


100000 


60 




1 


6.463726 




000000 




6.463726 




13.536274 


00029 


100000 


59 




2 


764756 




000000 




764756 




235244 


00058 


100000 


58 




3 


940847 




000000 




940847 




059153 


00087 


100000 


57 




4 


7.065786 




000000 




7.065786 




12.934214 


00116 


100000 


56 




5 


162696 




000000 




162696 




837304 


00145 


100000 


55 




6 


241877 




9.999999 




241878 




758122 


00175 


100000 


54 




7 


308824 




999999 




308825 




691175 


00204 


100000 


53 


8 


366816 




999999 




366817 




633183 


00233 


100000 


52 




9 


417968 




999999 




417970 




582030 


00262 


100000 


51 




10 


463725 




999998 




463727 




536273 


00291 


100000 


50 




11 


7.505118 




9.999998 




7.505120 




12.494880 


00320 


99999 


49 




12 


542905 




999997 




542909 




457091 


00349 


99999 


48 




1 13 


577668 




999997 




577672 




422328 


00378 


99999 


47 




I 14 


609853 




999996 




609857 




390143 


00407 


99999 


46 




15 


639816 




999996 




639820 




360180 


00436 


99999 


45 




16 


667845 




999995 




667849 




332151 


00465 


99999 


44 




17 


694173 




999995 




694179 




305821 


00495 


99999 


43 




18 


718997 




999994 




719003 




280997 


00524 


99999 


42 




19 


742477 




999993 




742484 




257516 


00553 


99998 


41 




20 


764754 




999993 




764761 




235239 


00582 


99998 


40 




21 


7.785943' 




9.999992 




7.785951 




12.214049 


00611 


99998 


39 




22 


808146 




999991 




806155 




193845 


00640 


99998 


38 




23 


825451 




999990 




825460 




174540 


00669 


99998 


37 




24 


843934 




999989 




843944 




156056 


00698 


99998 


36 




25 


861663 




999988 




861674 




138326 


00727 


99997 


35 




26 


878695 




999988 




878708 




121292 


00756 


99997 


34 




27 


895085 




999987 




895099 




104901 


00785 


99997 


33 




28 


910879 




999986 




910894 




089106 


00814 


99997 


32 


jj 


29 


926119 




999985 




926134 




073866 


00844 


99996 


31 


1 


30 


940842 




999983 




940858 




059142 


00873 


99996 


30 


\ 


31 


7.955082 


2298 
2227 
2161 


9.999982 


0.2 

0.2 
0.2 


7.955100 


2298 
2227 
2161 
2098 
2039 
1983 
1930 
1880 
1833 
1787 
1744 
1703 
1664 
1627 
1591 
1557 
1524 
1493 
1463 
1434 
1406 
1379 
1353 
1328 
1304 
1281 
1259 
1238 
1217 


12.044900 


00902 


99996 


29 


; 


32 


968870 


999981 


968889 


031111 


00931 


99996 


28 


jj 


33 


982233 


999980 


982263 


017747 


00960 


99995 


27 




34 


995198 


999979 


995219 


004781 


00989 


99995 


26 


1 


35 


S. 007787 


2098 
2039 


999977 


0-2 
0-2 


8.007809 


11.992191 


01018 


99995 


25 


\ 


36 


020021 


999976 


020045 


979955 


01047 


99995 


24 




37 


031919 


1983 
1930 
1880 
1832 
1787 
1744 
1703 
1664 
1626 
1591 
1557 
1524 
1492 
1462 
1433 
1405 
1379 
1353 
1328 
1304 
1281 
1259 
1237 
1216 


999975 


0-2 
0-2 
0-2 


031945 


968055 


01076 


99994 


23 




38 


043501 


999973 


043527 


956473 


01105 


99994 


22 




39 


054781 


999972 


054809 


945191 


01134 


99994 


21 




40 


085776 


999971 


0*2 
02 
- 2 
0*2 
0'2 
03 
0'3 
3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.4 
0.4 
0.4 
0.4 


065806 


934194 


01164 


99993 


20 




41 


8.076500 


9.999969 


8.076531 


11.923469 


01193 


99993 


19 




42 


086965 


999968 


086997 


913003 


01222 


99993 


18 




43 


097183 


999966 


097217 


902783 


01251 


99992 


17 




44 


107167 


999964 


107202 


892797 


01280 


99992 


16 




45 


116926 


999963 


116963 


883037 


01309 


99991 


15 




46 


126471 


999961 


126510 


873490 


01338 


99991 


14 




47 


135810 


999959 


135851 


864149 


01367 


99991 


13 




48 


144953 


999958 


144996 


855004 


01396 


99990 


12 




49 


153907 


999956 


153952 


846048 


01425 


99990 


11 




50 


162681 


999954 


162727 


837273 101454 


99989 


10 




51 


8.171280 


9.999952 


8.171328 


11.8286721101483 


99989 


9 




52 


179713 


999950 


179763 


820237 | 01513 


99989 


8 




53 


187985 


999948 


188036 


81 1964 J 101542 


99988 


7 




54 


196102 


999946 


196156 


803844 '01571 


99988 


6 




55 


204070 


999944 


204126 


795874 ; 01600 


99987 


6 




50 


211895 


999942 


211953 


788047|! 01629 


99987 


4 




57 


219581 


999940 


219641 


780359 '101658 


99986 


3 




58 


227134 


999938 


227195 


772805' '01687 


99986 


o 


59 


234557 


999936 


234621 


765379 01716 


99985 


1 


60 


241855 


999934 


241921 


758079 ,01745 


99985 







Cosine. 




Sine. 




Cotang. 




T&ng. N N. cos. 


N. sine- 


-J- 




89 Degrees. 









22 



Log. Sines and Tangents. (1°) Natural Si 



TABLE II. 



D.10" Cosine. D.10" 



Sine. 





1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
18 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



.241855 
249033 
256094 
263042 
269881 
276614 
283243 
289773 
296207 
302546 
308794 

.314954 
321027 
327016 
332924 
338753 
344504 
350181 
355783 
361315 
366777 

.372171 
377499 
382762 
387962 
393101 
398179 
403199 
408161 
413088 
417919 

.422717 
427462 
432156 
436800 
441394 
445941 
450440 
454893 
459301 
463665 

;. 467985 
472263 
476498 
480693 
484848 
488963 
493040 
497078 
501080 
505045 

1.508974 
512867 
516726 
520551 
524343 
528102 
531828 
535523 
539186 

J542819 

Cosine. 



1196 

1177 

1158 

1140 

1122 

1105 

1088 

1072 

1056 

1041 

1027 

1012 

998 

985 

971 

959 

946 

934 

922 

910 

899 

888 

877 

867 

856 

846 

837 

827 

818 

809 

800 

791 

782 

774 

766 

758 

750 

742 

735 

727 

720 

712 

706 

699 

692 

686 

679 

673 

667 

661 

655 

649 

643 

637 

632 

626 

i 621 

i 616 

| 611 

605 



9.999934 
999932 
999929 
999927 
999925 
999922 
999920 
999918 
999915 
999913 
999910 
999907 
999905 
999902 
999899 
999897 
999894 
999891 
999888 
999885 
999882 

9.999879 
999876 
999873 
999870 
999867 
999864 
999861 
999858 
999854 
999851 
999848 
999844 
999841 
999838 
999834 
999831 
999827 
999823 
999820 
999816 
999812 



999805 
999801 
999797 
999793 
999790 
999786 
999782 
999778 



0.4 
0-4 
0-4 
0.4 
0-4 
0.4 



0.5 
0.5 
0-5 
0-6 
0-6 
0.6 
0-6 
0-6 
0.6 
0.6 
0.6 
0-6 
0.6 
0-6 
0-6 



9.99977.-.' f X 
999769 U 
999765 
999761 
999757 
999753 
999748 
999744 
999740 
999735 



Si 



Tang. 



D.10" 



3.241921 
249102 
256165 
263115 
269956 
276691 
283323 
289856 
296292 
302634 
308884 

8.315046 
321122 
327114 
333025 
333856 
344610 
350289 
355895 
361430 
366895 

8.372292 
377622 
382889 
388092 
393234 
398315 
403338 
408304 
413213 
418088 

8.422869 
427618 
432315 
436962 
441560 
446110 
450613 
455070 
459481 
463849 

8.468172 
472454 
476693 
480892 
485050 
489170 
493250 
497293 
501298 
505267 

8.509200 
513098 
516961 
520790 
524586 
528349 
532080 
535779 
539447 
543084 



Cotaiu 



1197 

1177 

1158 

1140 

1122 

1105 

1089 

1073 

1057 

1042 

1027 

1013 

999 

985 

972 

959 

946 

934 

922 

911 



879 
867 
857 
847 
837 
828 
818 
809 
800 
791 
783 
774 
766 
758 
750 
743 
735 
728 
720 
713 
707 
700 
693 
686 
680 
674 
668 
661 
655 
650 
644 
638 
633 
627 
622 
616 
611 
606 



Coian< 



11.758079 
750898. 
743835 
736885 
730044 
723309 
716677 
710144 
703708 
697366 
691116 

11-684954 
678878 
672886 
666975 
661144 
655390 
649711 
644105 
638570 
633105 

11-627708 
622378 
617111 
611908 
606766 
601685 
596662 
591696 
586787 
581932 

11.577131 
572382 
567685 
563038 
558440 
653890 
549387 
544930 
540519 
536151 

11.531828 
527546 
523307 
619108 
514950 
510830 
506750 
502707 
498702 
494733 

11.490800 
486902 
483039 
479210 
476414 
471651 
467920 
464221 
460553 
456916 



N. sine. N. cos 



Tang. 



01742 

01774 

01803 

01832 

01862 

01891 

01920 

01949 

01978 

0200 

02036 

02065 

02094 

02123 

02152 

02181 

02211 

02240 

02269 

02298 

02327 

02356 

02385 

02414 

02443 

02472 

02501 

02530 

02560 

02589 

02618 

0264 

02676 

02705 

02734 

02763 

02792 

02821 

02850 

02879 

02908 

02938 

02967 

02996 

03025 

03054 

03083 

03112 

03141 

03170 

03199 

03228 

03257 

03286 

03316 

03345 

03374 

03403 

03432 

03461 

03490 



99985 60 
999841 59 



99984 58 
99983 57 
99983 56 
99982] 55 
99982 54 
99981 53 
99980; 62 
99980! 51 
99979J 50 
99979J 49 
y9978 48 
99977 47 
99977 46 
99976 45 
99976 44 
99975' 43 



99974! 42 



99974 

99973 

99972 

99972 

99971 

99970 

99969 

99969 

99968 

99967 

99966 

99966 

99965 

99984 

99963 

99963 

99962 

99961 

99960 

99959 

99959 

99958 

99957 

99956 

99955 

99954 

99953 

9995^ 

99952 

99951 

99950 

99949 

99948 

9994 

99946 

99945 

99944 

99943 

99942 

99941 

99940 

99939 



N. cos. N.sine 



83 Decrees. 



TABLE II. Log. Sines and Tangents. (2°) Natural Pines. 



23 



D. 10" i Cotang. || N. sine. lN. cos 



ine. 



018 

1 

2 

3 

4 

5 

6 

7 



60 



.542819 
545422 
549995 
553539 
557054 
560540 
563999 
567431 
570836 
574214 
577566 

.580892 
584193 
587469 
590721 
593948 
597152 
600332 
603489 
606623 
609734 

.612823 
615891 
618937 
621962 
624965 
627948 
630911 
633854 
636776 
639680 

.642563 
645428 
648274 
651102 
653911 
656702 
659475 
662230 
664968 
667689 

.670393 
673080 
675751 
678405 
681043 
683665 
686272 
688863 
691438 
693998 

.696543 
699073 
701589 
704090 
706577 
709049 
711507 
713952 
716383 
718800 



Cosmt 



D. 10" Cosine. 



600 
595 
591 
586 
581 
576 
572 
567 
563 
559 
554 
550 
546 
642 
538 
534 
530 
526 
522 
519 
516 
511 
508 
504 
501 
497 
494 
490 
487 
484 
481 
477 
474 
471 
468 
465 
462 
459 
456 
453 
451 
448 
445 
442 
440 
437 
434 
432 
429 
427 
424 
422 
419 
417 
414 
412 
410 
407 
405 
403 



.999735 
999731 
999726 
999722 
999717 
999713 
999708 
999704 
999699 
999694 
999689 

.999685 
999680 
999675 
999670 
999665 
999660 
999655 
999650 
999645 
999640 

.999635 
999629 
999324 
999619 
999614 



999603 
999597 
999592 



.999581 
999575 
999570 
999564 
999558 
999553 
999547 
999541 
999535 
999529 
.999524 
999518 
999512 
999506 
999500 
999493 
999487 
999481 
999475 
999469 
.999463 
999456 
999450 
999443 
999437 
999431 
999424 
999418 
999411 
999404 



Sine. 



D. 10" 



0.7 
0.7 
0-7 
0-8 
0-8 
0-8 
0.8 
0.8 
0-8 
0-8 
0.8 
0-8 
0-8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.9 
0.9 
0.9 
0.9 
0-9 
0-9 
0-9 
0-9 
0.9 
0.9 
0-9 
0.9 
0.9 
0.9 
0.9 
1-0 
10 
1.0 
1.0 



Tail' 



1.543084 
546691 
550268 
653817 
657335 
660828 
664291 
567727 
571137 
674520 
577877 

.581208 
584514 
587795 
591051 
594283 
697492 
600677 
603839 
606978 
610094 

.613189 
616262 
619313 
622343 
625352 
628340 
631308 
634256 
637184 
640093 

.642982 
645853 
648704 
651537 
654352 
657149 
659928 
662689 
665433 
668160 

.670870 
673563 
676239 
678900 
681544 
684172 
6S6784 
689381 
691963 
694529 

.697081 
699617 
702139 
704246 
707140 
709618 
702083 
714534 
716972 
719396 



Cot; 



602 
596 
591 
587 
582 
677 
673 
568 
664 
559 
555 
551 
647 
543 
639 
535 
531 
527 
523 
519 
616 
512 
508 
505 
501 
498 
495 
491 
488 
485 
482 
478 
475 
472 
469 
466 
463 
460 
457 
454 
453 
449 
446 
443 
442 
438 
435 
433 
430 
428 
425 
423 
420 
418 
415 
413 
411 
408 
406 
404 



11.456916 
453309 
449732 
446183 
442664 
439172 
435709 
432273 
428863 
425480 
422123 

11.418792 
415486 
412205 
408949 
405717 
402508 
399323 
396161 
393022 
389906 

11.386811 



03490 
03519 
03548 
03577 
03606 
03635 
.03664 
03693 
03723 
03752 
03 781 
03810 
03839 
03868 
03897 
03926 
03955 
03984 
04013 
04042 
04071 
04100 
03129 
04159 
04188 
04217 
04246 
04275 
04304 
04333 
04362 
04391 
04420 
04449 
04478 
04507 
04536 
04565 
04594 
04623 
04653 
04682 
04711 
04740 
04769 
04798 
04827 
04856 
04885 
04914 
04943 
04972 
05001 
05030 
05059 
05088 
05117 
05146 
05175 
05205 
05234 
Tan?. 1 1 N. cos. 



380687 
377657 
374648 
371660 
368692 
365744 
362816 
359907 

11.357018 
354147 
351296 
348463 
345648 
342851 
340072 
337311 
334567 
331840 

11.329130 
326437 
323761 
321100 
318456 
316828 
313216 
310619 
308037 
306471 

11.302919 
300383 
297861 
295354 
292860 
290382 
287917 
285465 
283028 
280604 



99939 60 



99938 
99937 
99936 
99935 
99934 
99933 
99932 
99931 
99930 
99929 
99927 
99926 
99925 
99924 
99923 
99922 
99921 
99919 
99918 
99917 
99916 
99915 
99913 
99912 
99911 
99910 
99909 
99907 
99906 
99905 
99904 
99902 
99901 
99900 



99897 



99894 
.99893 
99892 
99890 



99888 
99886 
99885 
99883 
99882 
99881 
79 
99878 
99876 
99875 
99873 
99872 
99870 
99869 
99867 
99866 
99864 
99863 



87 Degrees. 



24 



Log. Sines and Tangents. (3°j Natural Sines. TABLE II. 



9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



830749 
832607 
834456 
836297 
838130 



841774 
843585 



Cosine. 



9. 



998976 
998967 



1. 

1. 

1. 

1. 

1- 

1. 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 



Sine. D. 10" Cosine. D. 10 

.718800 
721204 
723595 
725972 
728337 
730688 
733027 
735354 
737667 
739969 
742259 
.744536 
746802 
749055 
751297 
753528 
755747 
757955 
760151 
762337 
764511 
.766675 
768828 
770970 
773101 
775223 
777333 
779434 
781524 
783605 
785675 
.787736 
789787 
791828 
793859 
795881 
797894 
799897 
801892 
803876 
805852 
.807819 
809777 
811726 
813667 
815599 
817522 
819436 
821343 
823240 
825130 
.827011 



401 
398 
396 
394 
392 
390 
388 
386 
384 
382 
380 
378 
376 
374 
372 
370 
368 
366 
364 
362 
361 
359 
357 
355 
353 
352 
350 
348 
347 
345 
343 
342 
340 
339 
337 
335 
334 
332 
331 
329 
328 
326 
325 
323 
322 
320 
319 
318 
316 
315 
313 
312 
311 
309 
308 
307 
306 
304 
303 
302 



999404 
999398 
999391 
999384 
999378 
999371 
999364 
999357 
999350 
999343 
999336 
999329 
999322 
999315 
999308 
999301 
999294 
999286 
999279 
999272 
999265 
999257 
999250 
999242 
999235 
999227 
999220 
999212 
999205 
999197 
999189 
999181 
999174 
999166 
999158 
999150 
999142 
999134 
999126 
999118 
999110 
999102 
999094 
999086 
999077 
999069 
999061 
999053 
999044 
999036 
999027 
999019 
999010 
999002 
998993 



998950 
998941 



1.2 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1-3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.5 
1.5 
1.5 



Tang. D. 10 



.719396 
721806 
724204 
726588 
728959 
731317 
733663 
735996 
738317 
740826 
742922 
.745207 
747479 
749740 
751989 
754227 
756453 
758668 
760872 
763085 
765246 
.767417 
769578 
771727 
773866 
775995 
778114 
780222 
782320 
784408 
786486 
.788554 
790613 
792662 
794701 
796731 
798752 
800763 
802765 
804858 
806742 
.808717 
810683 
812641 
814589 
816529 
818461 



822298 
824205 
826103 
.827992 
829874 
831748 



835471 
837321 
839163 
840998 
842825 
844644 



Cotanfr. 



402 
399 
397 
395 
393 
391 
389 
387 
385 
383 
381 
379 
377 
375 
373 
371 
369 
367 
365 
364 
362 
360 
358 
356 
355 
353 
351 
350 
348 
346 
345 
343 
341 
340 
338 
337 
335 
334 
332 
331 
329 
328 
326 
325 
323 
322 
320 
319 
318 
316 
315 
314 
312 
311 
310 
308 
307 
306 
304 
303 



Cotang. I |N. sine. 



11.25' 



11.280604 
278194 
275796 
273412 
271041 
268683 
266337 
264004 
261683 
259374 
257078 
254793 
252521 
250260 
248011 
245773 
243547 
241332 
239128 
236935 
234754 

11.232583 
230422 
228273 
226134 
224005 
221886 
219778 
217680 
215592 
213514 

11.211446 
209387 
207338 
205299 
203269 
201248 
199237 
197235 
195242 
193258 
191283 
189317 
187359 
185411 
183471 
181539 
179616 
177702 
175795 
173897 

11.172008 
170126 
168252 
166387 
164529 
162679 
160837 
159002 
157175 
155356 



05234 
05263 
05292 
05321 
05350 9985 



05379 
05408 
05437 
05466 
05495 
05524 
j 05553 
05582 
105611 
I 05640 
05669 
0i 
05727 
05756 
05785 
05814 
05844 
05873 
05902 
05931 
05960 
05989 
06018 
06047 
06076 
06105 
06134 
06163 
06192 
06221 
06250 
06279 



06337 
06366 
06395 
06424 
06453 
06482 
06511 
06540 
08569 
06598 
06627 
06656 
06685 
06714 
06743 
06773 
06802 
06831 
06860 
06889 
06918 
06947 
06976 



Tang. N. cos. X.sine 



99861 
99860 



99855 
99854 
99852 
99851 
99849 
99847 
99846 
99844 
99842 
99841 
99839 
99838 
99836 
99834 
99833 
99831 
99829 
99827 
99826 
99824 
99822 
99821 
99819 
9981 
9981& 
99813 
99812 
99810 
99808 
99806 
99804 
99803 
99801 
99799 
99797 
99795 
99793 
99792 
790 
788 
99786 
99784 
99782 
99780 
99778 
99776 
99774 
99772 
99770 
99768 
99766 
99764 
99762 
99760 
58 
99756 



86 Degrees. 



TABLE II. Log. Sines and Tangents. (4°) Natural Sines. 



25 



bine. 



1.843585 
845387 
847183 
848971 
850751 
852525 
854291 
856049 
857801 
859546 
861283 

.863014 
864738 
866455 
868165 
869868 
871565 
873255 
874938 
876615 
878285 

.879949 
881607 
883258 
884903 



888174 
889801 
891421 
893035 
894643 
.896246 
897842 
899432 
901017 
902596 
904169 
905736 
907297 
908853 
910404 
.911949 
913488 
915022 
916550 
918073 
919591 
921103 
922610 
924112 
925609 
.927100 
928587 
930068 
931544 
933015 
934481 
935942 
937398 
938850 
94u296 



Cosine. 



D. 10 



300 

299 

298 

297 

295 

294 

293 

292 

291 

290 

288 

287 

286 

285 

284 

283' 

282 

281 

279 

279 

277 

276 

275 

274 

273 

272 

271 

270 

269 

268 

267 

266 

265 

264 

263 

262 

261 

260 

259 

258 

257 

257 

256 

255 

254 

253 

252 

251 

250 

249 

249 

248 

247 

246 

245 

244 

243 

243 

242 

241 



Cosine. 



9.998941 
998932 
998923 
998914 
998905 
998896 
998887 
998878 
998869 
998860 
998851 



998832 



998813 
998804 
998795 
998785 
998776 
998766 
998757 
9.998747 
998738 
998728 
998718 
998708 
998699 
998689 
998679 
998669 
998659 



998639 
998629 
998619 
998609 
998599 
998589 
998578 
998568 
998558 

9.998548 
998537 
998527 
998516 
998506 
998495 
998485 
998474 
998464 
998453 

9.998442 
998431 
998421 
998410 



998388 
998377 
998366 
998355 
998344 



Sine. 



D. 10' 



Tang. 



.844644 
840455 
848260 
850057 
851846 
853628 
855403 
857171 
858932 
860686 
862433 
.864173 
865906 
867632 
869351 
871064 
872770 
874469 
876162 
877849 
879529 
.881202 
882869 
884530 
886185 
887833 
889476 
891112 
892742 
894366 
895984 
.897596 
899203 



902398 
903987 
905570 
907147 
908719 
910285 
911846 
.913401 
914951 
916495 
918034 
919568 
921096 
922619 
924136 
925649 
927156 
.928658 
930155 
931647 
933134 
934616 
936093 
937565 
939032 
940494 
941952 



Cotang. 



302 
301 
299 
298 
297 
296 
295 
293 
292 
291 
290 
289 
288 
287 
285 
284 
283 
282 
281 
280 
279 
278 
277 
276 
275 
274 
273 
272 
271 
270 
269 
268 
267 
266 
265 
264 
263 
262 
261 
260 
259 
268 
257 
256 
256 
255 
254 
253 
252 
251 
250 
249 
249 
248 
247 
246 
245 
244 
244 
243 



Cotang. j|N. sine. N. cos 

11.165356 

153545 

151740 

149943 

148154 

146372 

144597 

142829 

141068 

139314 

137567 
11.135827 

134094 

132368 

130649 

128936 

127230 

126531 

123838 

122151 

120471 
11.118798 

117131 

115470 

113815 

112167 

110524 



107258 
105634 
104016 
, 102404 
100797 
099197 
097602 
096013 
094430 
092853 
091281 
089715 
088154 

11.086599 
085049 
083505 
081966 
080432 
078904 
077381 
075864 
074361 
072844 

11.071342 
069845 
068353 
066866 
065384 
063907 
062435 
060968 
059506 
058048 



06976 
07005 
07034 
07063 
07092 
07121 
07150 
07179 
07208 
07237 
07266 
07295 
07324 
07353 
07382 
07411 
07440 
07469 
07498 
07527 
07556 
07585 



99756 
99764 
99752 
99750 
99748 
99746 
99744 
99742 
99740 
38 
99736 
99734 
99731 
99729 
99727 
99725 
99723 
99721 
99719 
99716 
99714 
99712 



0761499710 

07643 

07672 

07701 

07730 

07759 

07788 

07817 

07846 

07875 

07904 

07933 

07962 

07991 

08020 

08049 

08078 

08107 

08136 

08165 

08194 

08223 

08252 
I! 08281 

08310 
1 08339 

I j 08368 
! 08397 
j 08426 

08455 
! 1 08484 

08513 

,08542 

'08571 

108600 
1108629 
[ j 08658 

I I 08687 
1 1 08716 



Tang. | J N. cos. N.sine. ' 



99708 
99705 
99703 
99701 
99699 
99696 
99694 
99692 
99689 
99687 
99685 
99683 
99680 
99678 
99676 
99673 
99671 
99668 
99666 
99664 
99661 
99659 
99657 
99654 
99652 
99649 
99647 
99644 
99642 
99639 
99637 
99635 
99632 
99630 
99627 
99625 
99622 
99619 



85 Degrees. 



26 



Log. Sines and Tangents. (5°) Natural Sines. TABLE IT. 





1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 

i. 940296 
941738 
943174 
944606 
946034 
947456 
948874 
950287 
951696 
953100 
954499 

;. 955894 
957284 
958670 
960052 
981429 
962801 
984170 
965534 



D. 10"| Cosine. D. 10 



8. 



9. 



968249 
969600 
970947 
972289 
973628 
974962 
976293 
977619 
978941 
980259 
981573 
982883 
984189 
985491 
986789 
988083 
989374 
990660 
991943 
993222 
994497 
995768 
997036 
998299 
999560 
000816 
002069 
003318 
004563 
005805 
007044 
008278 
009510 
010737 
011962 
013182 
014400 
015613 
016824 
018031 I 
019235 I 



240 

239 

239 

238 

237 

236 

235 

235 

234 

233 

232 

232 

231 

230 

229 

229 

228 

227 

227 

226 

225 

224 

224 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

216 

216 

215 

214 

214 

213 

212 

212 

211 

211 

210 

209 

209 

208 

208 

207 

206 

206 

205 

205 

204 

203 

203 

202 

202 

201 

201 



9.998344 



998322 
998311 
998300 
998289 
998277 
998266 
998255 
998243 
998232 

.998220 
998209 
998197 
998186 
998174 
998163 
998151 
998139 
998128 
998116 

.998104 
998092 
998080 



998044 
998032 
998020 



997996 
.997984 
997972 
997959 
997947 
997935 
997922 
997910 
997897 
997885 
997872 
,997860 
997847 
997835 
997822 
997809 
997797 
997784 
997771 
997758 
997745 
9.997732 
997719 
997706 
997693 
997680 
997667 
997654 
997641 
997628 
997614 



Cosine. | 



1.9 
1.9 
1.9 
1.9 
1.9 



1 

1 

1 

1 

1 

1 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 



2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.1 
2.1 
2.1 



T ang. D. 10" Cotang. N. sine. W. cos 



2.1 
2.1 
2.1 
2.1 
2.1 



2.1 
2.1 

2.2 
2.2 
2.2 
2.2 
2.2 
2.2 



Sine. I 



8.941952 
943404 
944852 
946295 
947734 
949168 
950597 
952021 
953441 
954856 
956267 

8.957674 
959075 
960473 
961866 
963255 
964639 
966019 
967394 
968766 
970133 

8.971496 
972855 
974209 
975560 
976906 
978248 
979586 
980921 
982251 
983577 



986217 
987532 
988842 
990149 
991451 
992750 
994045 
995337 
996624 

8.997908 
999188 

9.000465 
001738 
003007 
004272 
005534 
006792 
008047 
009298 

9.010546 
011790 
013031 
014268 
015502 
016732 
017959 
019183 
020403 
021620 



CotaftH 



242 

241 

240 

240 

239 

238 

237 

237 

236 

235 

234 

234 

233 

232 

231 

231 

230 

229 

229 

228 

227 

226 

226 

225 

224 

224 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

216 

216 

215 

215 

214 

213 

213 

212 

211 

211 

210 

210 

209 

208 

208 

207 

207 

206 

206 

205 

204 

204 

203 

203 



11.058048 
056596 
055148 
053705 
052266 
050832 
049403 
047979 
046559 
045144 
043733 
11.042326 
040925 
039527 
038134 
036745 
035S61 
033981 
032606 
031234 
029867 
11.028504 
027145 
025791 
024440 
023094 
021752 
020414 
019079 
017749 
016423 
11.015101 
013783 
012468 
011158 
009851 
008549 
007250 
005955 
004663 j 
003376 
11.002092 
000812 
10.999535 1 
998262 | 
996993 | 
995728 I 
994466 | 
993208 | 
991953 
990702 
10.989454 
988210 
686969 
985732 j 
984498 I ! 



0871C 
08745 
08774 
08803 
08831 
08860 



08918 
08947 
08976 
09005 
09034 
09063 
1 09092 
09121 
09150 
09179 
09208 
09237 
09266 
09295 
09324 
09353 



99619 60 
99617 59 



58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
09556|99542l 31 



99614 

99612 

99609 

9960 

99604 

99602 

99599 

99596 

99594 

99591 

99588 

99586 

99583 

99580 

99578 

99575 

99572 

99570 

99567 

99564 

99562 



09382J99559 
09411 199556 
09440J99553 
09469199551 
09498J99548 
09527199545 



09585199540 
09614(99537 
09642J99534 
09671(99531 
09700J99528 
0972999526 
06758:99553 
09787|99520 
09816:99517 
09845J99514 



09874|99511| 20 



983041 
980817 
979597 
978380 I 



09903199508 

09932i9950:i 

09961 19950 

09990!99500 

1001919949 

10048J99494 

10077|99491 

10106I9948S 

10135J99485 

10164(99482 

10192 99479 

10221 [99476 

10-250,99473 

10279,99470 

10308 99467 



10337 
10366 
10395 



99464 
99461 
99458 



1042499455 

10453I9945-2 

Tang. |j N. cps.]\.pine. 



84 D^ree 



Log. Sines and Tangents. (6 C ) Natural Sines. 



27 



Sine. 



019235 
020435 
021632 
022825 
024016 
025203 
026386 
027567 
028744 
029918 
031089 
032257 
033421 
034582 
035741 
036896 
038048 
039197 
040342 
041485 
042625 

9.043762 
044895 
046026 
047154 
048279 
049400 
050519 
051635 
052749 
053859 

9.054966 
056071 
057172 
058271 
059367 
060460 
061551 
062639 
063724 
064806 
.065885 
066962 
068036 
069107 
070176 
071242 
072306 
073366 
074424 
075480 
.076533 
077583 
078631 
079676 
080719 
081759 
082797 
083832 
084864 
085894 
Cosine. 



18 



D. 10 




N. sine. N. cos. 



0453 
0482 
.0511 
0540 
0569 
0597 
0626 
0855 
0684 



99452 60 
99449 59 
99446 I 58 
99443 | 57 
99440 
99437 
99434 
99431 
99428 



0713 99424 



074-2 
0771 



99421 
99418 



0800 "9415 
0829 99412 



56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
99383 '' 38 
99380 | 37 
1147 99377136 
1176 y 9374 35 
1205 99370 34 
1234 99367|33 
1263 99364 32 



0858 
0887 
0916 
0945 
0973 
1002 
1031 
1080 
1089 
1118 



99409 
99406 
99402 
99399 
99396 
99393 
99390 



1291 99360 
1320 99357 
1349 99354 
99351 
99347 
99344 
99341 
99337 
99334 
99331 
99327 
99324 
99320 
99317 
99314 
99310 
99307 
99303 
99300 
99297 
99293 
99290 
99286 
99283 



1378 
1407 
1436 
1465 
1494 
1523 
1552 
1580 
1609 
1638 
1667 
1696 
1725 
1754 
1783 
1812 
1840 
1869 
1898 
1927 
1956 



1985 99279 
2014 99276 
.2043 99272 
2071 99269 
2100 99265 
2129 99262 
158 99258 
2187 99255 



I N. eos.|N.sine 



Degrees. 



28 



Log. Sines and Tangents. (7°) Natural Sines. 



Sine. 



Tans. 



WW 



N. sine 



WW 



9.085894 
086922 
087947 
088970 
089990 
091008 
092024 
093037 
094047 
095056 
096062 

9.097065 
098066 
099065 
100062 
101056 
102048 
103037 
104025 
105010 
105992 
.106973 
107951 
108927 
109901 
110873 
111842 
112809 
113774 
114737 
115698 
.116656 
117613 
118567 
119519 
120469 
121417 
122362 
123306 
124248 
125187 

9.126125 
127060 
127993 
128925 
129854 
130781 
131706 
132630 
133551 
134470 

9.135387 
136303 
137216 
138128 
139037 
139944 
140850 
141754 
142655 
143555 



Cosine. 



171 
171 
170 
170 
170 
169 
169 
168 
168 
168 
167 
167 
166 
166 
166 
165 
165 
164 
164 
164 
163 
163 
163 
162 
162 
162 
161 
161 
160 
160 
160 
159 
169 
159 
158 
158 
158 
157 
157 
157 
156 
156 
156 
155 
155 
154 
154 
154 
153 
153 
153 
152 
152 
152 
152 
151 
151 
151 
150 
150 



Cosine. 



D. 10" 



.996751 
996735 
996720 
996704 
996688 
996673 
996657 
996641 
996625 
996610 
996594 

.996578 
996562 
996546 
996530 
996514 
996498 
996482 
996465 
996449 
996433 

.996417 
996400 
996384 
996368 
996351 
996335 
996318 
996302 
996285 



.996252 
996235 
996219 
996202 
996185 
996168 
996151 
996134 
996117 
996100 



996066 
996049 
996032 
996015 
995998 
995980 
995963 
995946 
995928 
.995911 
995894 
995876 
995859 
995841 
995823 
995806 
995788 
995771 
995753 



Sine. 



2.6 

2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 



2.7 
2.7 
2.7 
2.7 

2.8 

2.8 

2.8 

2.8 

2. 

2, 

2, 

2, 

2. 

2. 

2. 

2. 

2. 



2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 



2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 



9. 



.089144 
090187 
091228 
092266 
093302 
094336 
095367 
096395 
097422 
098446 



100487 
101504 
102519 
103532 
104542 
105550 
106556 
107559 
108560 
109559 
110556 
111551 
112543 
113533 
114521 
115507 
116491 
117472 
118462 
119429 
120404 
121377 
122348 
123317 
124284 
125249 
126211 
127172 
128130 
129087 
130041 
130994 
131944 
132893 
133839 
134784 
135726 
136667 
137605 
138542 
139476 
140409 
141340 
142269 
143196 
144121 
145044 
145966 
146885 
147803 



Cotang. 



174 
173 
173 
173 
172 
172 
171 
171 
171 
170 
170 
169 
169 
169 
168 
168 
168 
167 
167 
166 
166 
166 
165 
165 
165 
164 
164 
164 
163 
163 
162 
162 
162 
161 
161 
161 
160 
160 
160 
159 
159 
159 
158 
168 
158 
157 
157 
157 
156 
156 
156 
155 
155 
155 
154 
154 
154 
153 
153 
153 



Uotang. 



10 



10 



10 



10 



10 



10 



.910856 
909813 
908772 
907734 
906698 
905664 
904633 
903605 
902578 
901554 
900532 

.899513 
898496 
897481 
896468 
895458 
894450 
893444 
892441 
891440 
890441 

.889444 
888449 
887457 
886467 
885479 
884493 
883509 
882628 
881548 
880571 

.879596 
878623 
877652 
876683 
875716 
874751 
873789 
872828 
871870 
870913 

.869959 
869006 
868056 
867107 
866161 
865216 
864274 
863333 
862395 
861458 

.860524 
859591 
858660 
857731 
856804 
855879 
854956 
854034 
853115 
852197 



13917 



99027 



Tang. I N. cos. N .sine 



, COH. 



12187 

12216 

12245 

12274 

12302 

12331 

12360 

12389 

12418 

12447 

12476 

12504 

12533 

12562 

12591 

12620 

12649 

12678 

12706 

12735 

12764 

12793 

12822 

12851 

12880 

12908 

12937 

12966 

12995 

13024 

13053 

13081 

13110 

13139 

13168 

13197 

13226 

13254 991 

13283 

13312 

13341 

13370 

13399 

13427 

13456 

13485 

13514 

13543 

13572 

13600 

13629 

13658 

13687 

13716 

13744 

13773 

13802 

13831 

13860 

13889 99031 



99255 
99251 
99248 
99244 
99240 
99237 
99233 
99230 
99226 
99222 
99219 
99215 
99211 
99208 
99204 
99200 
99197 
99193 
99189 
99186 
99182 
99178 
99175 
99171 
99167 
99163 
99160 
99156 
99152 
99148 
99144 
99141 
99137 
99133 
99129 
99125 
99122 
18 
99114 
99110 
99106 
99102 
99098 
99094 
99091 
99087 
99083 
99079 
99075 
99071 
99067 
99063 
99059 
99055 
99051 
99047 
99043 
99039 
99035 



82 Degrees. 



Log. Sines and Tangents. (8°) Natural Sines 



29 



Sine. 





1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

1? 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 



35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



. 143555 
144453 
145349 
146243 
147136 
148026 
148915 
149802 
150686 
151569 
152451 

.153330 
154208 
155083 
155957 
156830 
157700 
158569 
159435 
160301 
161164 

. 162025 
162885 
163743 
164600 
165454 
166307 
167159 
168008 
168856 
169702 

.170547 
171389 
172230 
173070 
173908 
174744 
175578 
176411 
177242 
178072 

.178900 
179726 
180551 
181374 
182196 
183016 
183834 
184651 
185466 
186280 

.187092 
187903 
188712 
189519 
190325 
191130 
191933 
192734 
193534 
194332 



Cosine. 



D. 10" 



150 
149 
149 
149 
148 
148 
148 
147 
147 
147 
147 
146 
146 
146 
145 
145 
145 
144 
144 
144 
144 
143 
143 
143 
142 
142 
142 
142 
141 
141 
141 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
137 
137 
137 
137 
136 
136 
136 
136 
135 
135 
135 
135 
134 
134 
134 
134 
133 
133 



Cosine. 



.995753 
995735 
995717 
995699 
995681 
995664 
995646 
995628 
995610 
995591 
995573 
.995555 
995537 
995519 
995501 
995482 
995464 
995446 
995427 
995409 
995390 
.995372 
995353 
995334 
995316 
995297 
995278 
995260 
995241 
995222 
995203 
.995184 
995165 
995146 
995127 
995108 
995089 
995070 
995051 
995032 
995013 
.994993 
994974 
994955 
994935 
994916 
994896 
994877 
994857 
994838 
994818 
.994798 
994779 
994759 
994739 
994719 
994700 
994686 
994660 
994640 
994620 



D. 10'' 



3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
8.0 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 



Tang. 



D. 10' 



). 147803 
148718 
149632 
150544 
151454 
152363 
153269 
154174 
155077 
155978 
156877 

M57775 
158671 
159565 
160457 
161347 
162236 
163123 
164008 
164892 
165774 

>. 166654 
167532 
168409 
169284 
170157 
171029 
171899 
172767 
173634 
174499 

1. 175362 
176224 
177084 
177942 
178799 
179655 
180508 
181360 
182211 
183059 

1.183907 
184752 
185597 
186439 
187280 
188120 
188958 
189794 
190629 
191462 

M 92294 
193124 
193953 
194780 
195606 
196430 
197263 
198074 
198894 
1 99713 

Cotang. 



153 
152 
152 
152 
151 
151 
151 
150 
150 
150 
150 
149 
149 
149 
148 
148 
148 
148 
147 
147 
147 
146 
146 
146 
145 
145 
145 
145 
144 
144 
144 
144 
143 
143 
143 
142 
142 
142 
142 
141 
141 
141 
141 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
138 
137 
137 
137 
137 
136 



Cotang. IjN. sine. N. cos, 



10.852197 
851282 
850368 
849456 
848546 
847637 
846731 
845826 
844923 
844022 
843123 

10. .842225 
841329 
840435 
839543 



837764 
836877 
835992 
835108 
834226 

10.833346 
832468 
831591 
830716 
829843 
828971 
828101 
827233 
826366 
825501 

10.824638 
823776 
822916 
822058 
821201 
820345 
819492 
818640 
817789 
816941 

10-816093 
815248 
814403 
813561 
812720 
811880 
811042 
810206 
809371 
808538 

10.807706 
806876 
806047 
805220 
804394 
803570 
802747 
801926 
801106 
800287 



* Tang. 



13917 
13946 
13975 
14004 
14033 
14061 
14090 
14119 
14148 
14177 
14205 
14234 
14263 
14292 
14320 
14349 
14378 
14407 
14436 
14464 
14493 
14522 
14561 
14580 
14608 
14637 
14666 
14695 
14723 
14752 
14781 
14810 
14838 
14867 
14896 
14926 
14954 
14982 
15011 
15040 
15069 
15097 
15126 
15155 
15184 
15212 
15241 
15270 
15299 
15327 
15356 
15385 
16414 
15442 
15471 
15500 
15529 
15557 
15586 
16615 
16643 



99027 
99023 
99019 
99015 
99011 
99006 
99002 
98998 
98994 
98990 
98986 
989S2 
98978 
)73 
98969 
98965 
98961 
98957 
98953 
98948 
98944 
98940 
98936 
98931 
98927 
98923 
98919 
98914 
98910 
98906 
98902 
98897 
98893 



98876 
98871 
98867 
98863 
98858 
98854 
98849 



98841 
98836 
98832 
98827 
98823 
98818 
98814 



98805 



98796 
98791 
98787 
98782 
98778 
98773 
98769 
N. cos. N.siue 



30 



Log. Sines and Tangents. (9°) Natural Sines. 



Sine. 



9.194332 
195129 
195925 
196719 
197511 
198302 
199091 
199879 
200666 
201451 
202234 

9.203017 
203797 
204577 
205354 
206131 
206906 
207679 
208452 
209222 
209992 

9.210760 
211526 
212291 
213055 
213818 
214579 
215338 
216097 
216854 
217609 

9.218363 
219116 
219868 
220618 
221367 
222115 
222861 
223606 
224349 
225092 

9.225833 
226573 
227311 
228048 
228784 
229518 
230252 
230984 
231714 
232444 

9.233172 
233899 
234625 
235349 
236073 
236795 
237515 
238235 
238953 
239670 



D. 10" Cosine. 



Cosine. 



133 
133 
132 
182 
132 
132 
131 
131 
131 
131 
130 
130 
130 
130 
129 
129 
129 
129 
128 
128 
128 
128 
127 
127 
127 
127 
127 
126 
126 
126 
126 
125 
125 
125 
125 
125 
124 
124 
124 
124 
123 
123 
123 
123 
123 
122 
122 
122 
122 
122 
121 
121 
121 
121 
120 
120 
120 
120 
120 
119 



994620 
994600 
994580 
994560 
994540 
994519 
994499 
994479 
994459 
994438 
994418 
994397 
994377 
994357 
994336 
994316 
994295 
994274 
994254 
994233 
994212 
9.994191 
994171 
994150 
994129 
994108 
994087 
994066 
994045 
994024 
994003 



D. 10" 



993960 



993918 



993875 
993854 
993832 
993811 
993789 
993768 
993746 
993725 
993703 
993681 
993660 
993638 
993616 
993594 
993572 
9.993550 
994528 
993506 
993484 
993462 
993440 
993418 
993396 
993374 
993351 



Sine. 



3.3 
3.3 
3.3 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 



3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.6 
3.5 
3.5 
3.6 
3.6 
3.5 
3.5 
3.5 
3.6 
3.5 
3.5 
3.6 



3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.7 
3.7 
3.7 



3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 



Tans. 



1.199713 
200529 
201345 
202159 
202971 
203782 
204592 
205400 
206207 
207013 
207817 

1.208619 
209420 
210220 
211018 
211815 
212611 
213405 
214198 
214989 
215780 

.216568 
217356 
218142 
218926 
219710 
220492 
221272 
222052 
222830 
223608 

.224382 
225166 
225929 
226700 
227471 
228239 
229007 
229773 
230539 
231302 

.232065 
232826 
233586 
234345 
235103 
235859 
236614 
237368 
238120 
238872 

,239622 
240371 
241118 
241865 
242610 
243354 
244097 
244839 
245579 
246319 



Cotang. 



D. 10" 

136 
136 
136 
135 
135 
135 
135 
134 
134 
134 
134 
133 
133 
133 
133 
133 
132 
132 
132 
132 
131 
131 
131 
131 
130 
130 
130 
130 
130 
129 
129 
129 
129 
129 
128 
128 
128 
128 
127 
127 
127 
127 
127 
126 
126 
126 
126 
126 
125 
125 
125 
125 
125 
124 
124 
124 
124 
124 
123 
123 



Cotany;. 



10.800287 
799471 
798655 
797841 
797029 
796218 
795408 
794600 
793793 
792987 
792183 

10.791381 
790580 
789780 



N. sine. N. co: 



788185 
787389 
786595 
785802 
735011 
784220 

10.783432 
782644 
781858 
781074 
780290 
779508 
778728 
777948 
777170 
776394 

10.775618 
774844 
774071 
773300 
772529 
771761 
770993 
770227 
769461 
768698 

10.767935 
767174 
766414 
765655 
764897 
764141 
763386 
762632 
761880 
761128 

10.760378 
759629 
758882 
758135 
757390 
756646 
755903 
755161 
754421 
753681 



15643 
15672 
15701 
15730 
15758 
15787 
15816 
15845 
15873 
15902 
15931 
15959 
15988 
16017 
16046 
16074 
16103 
16132 
16160 
| 16189 
16218 
16246 
16275 
16304 
16333 
16361 
16390 
16419 
16447 
16476 
16505 
16533 
16562 
16591 
16620 
16648 
16677 
16706 
1116734 
H16763 
1 1 16792 
16820 
16849 
1687 
16906 
16935 
16964 
16992 
17021 
17050 
17078 
17107 
17136 
17164 
17193 
17222 
17250 
17279 
17308 
17336 
17365 



Tang. li N. cos. N.sine. 



764 

98760 
98755 
98751 
98746 
98741 

737 
98732 
98728 
98723 
98718 
98714 

709 
98704 
98700 
98695 
98690 
98686 
98681 
98676 
98671 
98667 
98662 
98657 
98652 
98648 
98643 
98638 
98633 
98629 



98624 | 29 



98619 
98614 
98609 
98604 
98600 
98595 
98590 
98585 
98580 
98575 
98570 
98565 
98561 
98556 
98551 
98546 
98541 
98536 
98531 
98526 
98521 
98516 
98511 
98506 
98501 
98496 
98491 
98486 
98481 



28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 




80 Degrees. 



TABLE II. 



Log. Sines and Tangents. (10°) Natural Sines. 



•31 



Sine. 

9.239670 
240386 
241101 
241814 
242526 
243237 
243947 
244656 
245363 
246089 
246775 

9.247478 
248181 
248883 
249583 
250282 
250980 
251677 
252373 
253067 
253761 

9.254453 
255144 
255834 
256523 
257211 
257898 
258583 
259268 
259951 
260633 

9.261314 
261994 
262673 
263351 
264027 
264703 
265377 
266051 
266723 
267395 

9.268065 
268734 
269402 
270069 
270735 
271400 
272064 
272726 
273388 
274049 

9.274708 
275367 
276024 
276681 
277337 
277991 
278644 
279297 
279948 
280599 
Cosine. 



D. 10" 



119 
119 
119 
119 
118 
118 
118 
118 
118 
117 
117 
117 
117 
117 
116 
116 
116 
116 
116 
116 
115 
115 
115 
115 
115 
114 
114 
114 
114 
114 
113 
113 
113 
113 
113 
113 
112 
112 
112 
112 
112 
111 
111 
111 
111 
111 
111 
110 
110 
110 
110 
110 
110 
109 
109 
109 
109 
109 
109 
108 



Cosine. 

'.993351 
993329 
993307 
993285 
993262 
993240 
993217 
993195 
993172 
993149 
993127 

'.993104 
993081 
993059 
993036 
993013 
992990 
992967 
992944 
992921 
992898 

'•992875 
992852 
992829 
992806 
992783 
992759 
992736 
992713 
992690 
992666 

.992643 
992619 
992596 
992572 
992549 
992525 
992501 
992478 
992454 
992430 

'.992406 
992382 
992359 
992335 
992311 
992287 
992263 
992239 
992214 
992190 

'.992166 
992142 
992117 
992093 
992069 
992044 
992020 
991996 
991971 
991947 
Siut\ 



D. 10"| 



3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.8 



3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.1 
4.1 



T ang. 

.246319 
247057 
247794 
248530 
249264 
249998 
250730 
251461 
252191 
252920 
253648 

.254374 
265100 
255824 
256547 
257269 
257990 
258710 
259429 
260146 
260863 

.261578 
262292 
263005 
263717 
264428 
265138 
265847 
266555 
267261 
267967 

.268671 
269375 
270077 
270779 
271479 
272178 
272876 
273573 
274269 
274964 

.275658 
276351 
277043 
277734 
278424 
279113 
279801 
280488 
281174 
281858 

. 282542 
283225 
283907 
284588 
285268 
285947 
286624 
287301 
287977 
288652 



Cotang. 



D. 10" 



123 
123 
123 
122 
122 
122 
122 
122 
121 
121 
121 
121 
121 
120 
120 
120 
120 
120 
120 
119 
119 
119 
119 
119 
118 
118 
118 
118 
118 
118 
117 
117 
117 
117 
117 
116 
116 
116 
116 
116 
116 
115 
115 
115 
115 
115 
115 
114 
114 
114 
114 
114 
114 
113 
113 
113 
113 
113 
113 
112 



Cotang. 

10.753681 
752943 
752206 
751470 
750736 
750002 
749270 
748539 
747809 
747080 
746352 

10.745626 
744900 
744176 
743453 
742731 
742010 
741290 
740571 
739854 
739137 

10.738422 
737708 
736995 
736283 
735572 
734862 
734153 
733445 
732739 
732033 

10.731329 
730625 
729923 
729221 
728521 
727822 
727124 
726427 
725731 
725036 

10.724342 
723649 
722957 
722266 
721576 
720887 
720199 
719512 
718826 
718142 

10.717458 
716775 
716093 
715412 
714732 
714053 
713376 
712699 
712023 
711348 



N.sine. N. cos. 

98481 
98476 
98471 
98466 
98461 
98455 
98450 
98445 
98440 
98435 
98430 
98425 
98420 
98414 
98409 
98404 
98399 
98394 
98389 
98383 
98378 
98373 



Tang. 



17365 
17393 
17422 
17451 
17479 
17508 
17537 
17565 
17594 
17623 
17651 
17680 
17708 
17737 
17766 
17794 
17823 
17852 
17880 
17909 
17937 
17966 
17995 
18023 
18052 
18081 
18109 
18138 
18166 
18195 
18224 
18252 
18281 
18309 
18338 
18367 
18395 
18424 
18452 
18481 
18509 
18538 
18567 
18595 
18624 
18652 
18681 
18710 
18738 
18767 
18795 
18824 
18852 
18881 
18910 
18938 
18967 
18995 
19024 



98362 
98357 
98352 
98347 
98341 
98336 
98331 
98325 
98320 
98315 
98310 
98304 
98299 
98294 



98283 
98277 
98272 
98267 
98261 
98256 
98250 
98245 
98240 
98234 
98229 
98223 
98218 
98212 
98207 
98201 
98196 
98190 
98186 
98179 
98174 



19052 98168 
19081 98163 



N. cos. N.sine, 



79 Degrees. 



32 



Log. Sines and Tangents. (11°) Natural Sines. 



TABLE II. 



Sine. 



D. 10' 





1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



'.294029 
294658 
295286 
295913 
296539 
297164 
297788 
298412 
299034 
299655 

'.300276 
300895 
301514 
302132 
302748 
303364 
303979 
304593 
305207 
305819 

'.306430 
307041 
307650 
308259 
308867 
309474 
310080 
310685 
311289 
311893 

'.312495 
313097 
313698 
314297 
314897 
315495 
316092 
316689 
317284 
317879 



Cosine. 



108 
108 
108 
108 
108 
107 
107 
107 
107 
107 
107 
106 
106 
106 
106 
106 
106 
105 
105 
105 
105 
105 
105 
104 
104 
104 
104 
104 
104 
104 
103 
103 
103 
103 
103 
103 
102 
102 
102 
102 
102 
102 
102 
101 
101 
101 
101 
101 
101 
100 
100 
100 
100 
100 
100 
100 
100 
99 
99 
99 



Cosine. 

.991947 
991922 
991897 
991873 
991848 
991823 
991799 
991774 
991749 
991724 
991699 

L991674 
991649 
991624 
991599 
991574 
991549 
991524 
991498 
991473 
991448 

.991422 
991397 
991372 
991346 
991321 
991295 
991270 
991244 
991218 
991193 

.991167 
991141 
991115 
991090 
991064 
991038 
991012 



990960 
990934 
990908 



990855 
990829 
990803 
990777 
990750 
990724 
990697 
990671 
.990644 
990618 
990591 
990565 
990538 
990511 
990485 
990458 
990431 
990404 



Sine. 



D. 10" 



4.1 
4.1 
4.1 



4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.5 
4.5 
4.5 
4.5 



Tanjj 



.288652 
289326 
289999 
290671 
291342 
292013 
292682 
293350 
294017 
294684 
295349 

.296013 
296677 
297339 
298001 
298662 
299322 
299980 
300638 
301295 
301951 

.302607 
303261 
303914 
304567 
305218 
305869 
306519 
307168 
307815 



D. 10' 



.309109 
309754 
310398 
311042 
311685 
312327 
312967 
313608 
314247 
314885 

.315523 
316159 
316795 
317430 
318064 
318697 
319329 
319961 
320592 
321222 

.321851 
322479 
323106 
323733 
324368 
324983 
325607 
326231 
326853 
327475 



Co tang. 



112 

112 

112 

112 

112 

11 

11 

11 

11 

11 

11 

11 

110 

110 

110 

110 

110 

110 

109 

109 

109 

109 

109 

109 

109 

108 

108 

108 

108 

108 

108 

107 

107 

107 

107 

107 

107 

107 

106 

106 

106 

106 

108 

106 

106 

105 

105 

105 

105 

105 

105 

105 

104 

104 

104 

104 

104 

104 

104 

104 



Cor.ang 



10.711348 
710674 
710001 
709329 
708658 
707987 
707318 
706650 
705983 
705316 
704651 

10.703987 
703323 
702661 
701999 
701338 
700678 
700020 
699362 
698705 
698049 

10-697393 
696739 
696086 
695433 
694782 
694131 
693481 
692832 
692185 
691537 

10-690891 
690246 
689602 
688958 
688315 
687673 
687033 
686392 
685753 
685115 

10-684477 
683841 
683205 
682570 
681936 
681303 
680671 
680039 
679408 
678778 

10.678149 
677621 



N. sine. N. cos 



676267 
675642 
675017 
674393 
673769 
673147 
672525 



Tan' 



19081 
19109 
19138 
19167 
19195 
19224 
19252 
19281 
19309 
19338 



19396 

19423 

19452 

19481 

19509 

19538 

19566 

19595 

19623 

19652 

19680 

19709 

19737 

19766 98027 

19794 98021 
98016 
98010 

19880 98004 

19908 9' 



19851 



7998 
97992 
97987 
97981 
97975 
97969 
97963 
97958 
97952 
97946 
97940 
97934 
97928 
97922 
97916 
97910 
97905 

7899 
97893 
97887 
97881 
97875 
97869 
97863 
97857 
97851 

7845 

7839 
97833 
97827 
97821 
97815 
I N. cos. N.Pino. 



19937 
19965 
19994 
20022 
20051 
20079 
20108 
20136 
20165 
20193 
20222 
20250 
20279 
20307 
20336 
20364 
20393 
20421 
20450 
20478 
20507 
20535 
20563 
20592 
20620 
20649 
20677 
20706 
20734 
20763 
20791 



98163 
98157 
98152 
98146 
98140 
98135 
98129 
98124 
98118 
98112 
98107 
98101 
98096 
98090 
98084 
98079 
98073 
98067 
98061 
98056 
98050 
98044 
98039 



18 Degrees. 



Log. Sines and Tangents. (12°) Natural Sines. 



33 



Sine. D. 10" Cosine. D. 10" Tang. 



9 
10 
11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



.317879 
318473 
319066 
319658 
320249 
320840 
321430 
322019 
322607 
323194 
323780 

.324366 
324950 
325534 
326117 
326700 
327281 
327862 
328442 
329021 
329599 

.330176 
330753 
331329 
331903 
332478 
333051 
333624 
334195 
334766 
335337 

.335908 
836475 
337043 
337610 
338176 
338742 
339306 
339871 
340434 
340996 

.341558 
342119 
342679 
343239 
343797 
344355 
344912 
345469 
346024 
346579 

.347134 
347687 
348240 
348792 
349343 
349893 
350443 
350992 
351540 
352088 



Cosine. 



99.0 

98.8 

98.7 

98.6 

98.4 

98.3 

98.2 

98.0 

97.9 

97.7 

97.6 

97.5 

97.3 

97.2 

97.0 

96.9 

96.8 

98.6 

98.5 

96.4 

98.2 

96.1 

96.0 

95.8 

95.7 

95.6 

95.4 

95.3 

95.2 

95.0 

94.9 

94.8 

94.6 

94 

94 

94 

94 

94 

93 

93 

93 

93 

93 

93.2 

93.1 

93.0 

92.9 

92.7 

92.6 

92.6 

92.4 

92.2 

92.1 

92.0 

91.9 

91.7 

91.6 

91.5 

91.4 

91.3 



989356 



4.5 
4.5 



4.6 
4.6 



4 

4 

4 

4 

4 

4 

4 

4 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 



4.9 
4.9 



D. 10" 



327474 
328095 
328715 
329334 
329953 
330570 
331187 
331803 
332418 
333033 
333646 

9.334259 
334871 
335482 
336093 
336702 
337311 
337919 
338527 
339133 
339739 

9.340344 
340948 
341552 
342155 
342757 
343358 
343958 
344558 
345157 
345755 

9.346353 
346949 
347545 
348141 
348735 
349329 
349922 
350514 
351106 
351697 

9.362287 
352876 
353465 
354053 
354640 
355227 
356813 
356398 
356982 
357566 

9.358149 
358731 
359313 
359893 
360474 
361053 
361632 
362210 
362787 
363364 
Cotang. 



103 

103 

103 

103 

103 

103 

103 

102 

102 

102 

102 

102 

102 

102 

102 

101 

101 

101 

101 

101 

101 

101 

101 

100 

100 

100 

100 

100 

100 

100 

100 
99.4 
99.3 
99.2 
99.1 
99.0 
98.8 
98.7 
98.6 
98.5 
98.3 
98.2 
98.1 
98.0 
97.9 
97.7 
97.6 
97 

97.4 
97.3 
97 

97.0 
96.9 
96.8 
96 
96 
96 
96 



10 



10 



Cotang. 

.672526 
671905 
671285 
670666 
670047 
669430 
668813 
668197 
667582 
666967 
666354 
.665741 
665129 
664518 
663907 
663298 
662689 
662081 
661473 
660867 
660261 
.659656 
659052 
658448 
657845 
657243 
656642 
656042 
655442 
654843 
654245 
.653647 
653051 
652455 
651859 
651265 
650671 
650078 
649486 
648894 
648303 
.647713 
647124 
646535 
645947 
646360 
644773 
644187 
643602 
643018 
642434 
.641851 
641269 
640687 
640107 
639526 
638947 
638368 
637790 
637213 
636636 



10 



N. sine.(iS T . cos. 



10 



10 



20791 97815 
20820 97809 
20848 97803 
20877 97797 
20905 97791 
20933 97784 
20962 97778 
20990 97772 
2101997766 
21047 97760 
21076 97754 
21104 9774S 
21132 97742 
2116197736 
21189 97729 
2121897723 
2124697717 
21275 97711 
21303 97705 
21331:97698 
21360 97692 
21388 97686 
2141797680 
2144597673 
21474197667 
21502 97661 
21530 97655 
21559!97648 
21587!97642 
21616J97636 
2164497630 
21672J97623 
2170197617 
21729 97611 
21768 97604 
21786 97598 
21814:97692 



Tang. 



21843 
21871 
21899 
21928 
21956 
21985 
22013 
22041 
22070 
22098 
22126 
22155 
22183 
22212 
22240 
22268 
22297 
22325 
22353 
22382 
22410 
22438 
22467 
22495 



97585 
97679 
97573 
97566 
97560 
97553 
97547 
97541 
97534 
97528 
97521 
97615 
97608 
97502 
97496 
97489 
97483 
97476 
97470 
97463 
97467 
97450 
97444 
97437 



N. cos. X.sinr. 



n Degrees. 



34 



Log. Sines and Tangents. (13°) Natural Sines. 



TABLE IL 



Sine. b. 10" Cosine. D. 10" Tancr. D. 10" Cotan 





1 

2 

3 

4 
5 
6 
7 
8 
9 

10 
11 
IS 
13 
14 
15 
16 
17 
18 
19 
20 
21 
2-2 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
5i 
53 
54 
55 
56 
57 
58 
59 
60 



.352088 
352635 
353181 
353726 
354271 
354815 
355358 
355901 
356443 
356984 
357524 

.358064 
358603 
359141 
359678 
360215 
360752 
361287 
361822 
362356 
362889 

.363422 
363954 
364485 
365016 
365546 
366075 
366604 
367131 
367659 
368185 

.368711 
369236 
369761 
370285 
370808 
371330 
371852 
372373 
372894 
373414 

.373933 
374452 
374970 
375487 
376003 
376519 
377035 
377549 
378063 
378577 

.379089 
379601 
380113 
380624 
381134 
381643 
382152 
382661 
383168 
383676 

Cosine. 



91.1 
91.0 
90.9 
90.8 
90.7 
90.5 
90.4 
90.3 
90.2 
90.1 
89.9 
89.8 
89.7 
89.6 
89.5 
89.3 
89.2 
89.1 
89.0 
88.9 



88.7 
88.5 
88.4 
88.3 
88.2 
88.1 
88.0 
87.9 
87.7 
87.6 
87.5 
87.4 
87.3 
87.2 
87.1 
87.0 
86.9 
86.7 
86.6 
86.5 
86.4 
86.3 
86.2 
86.1 
86.0 
85.9 
85.8 
85.7 
85.6 
85.4 
85.3 
85.2 
85.1 
85.0 
84.9 
84.8 
84.7 
84.6 
84.5 



Sine. 



4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
5.0 
6.0 
6.0 
5.0 
5.0 



6.1 
5.1 
6.1 
5.1 
5.1 
6.1 



6.1 

6.2 
5.2 

5.2 

6.2 

5.2 

5.2 

6.2 

5.2 

2 

2 

2 

2 

2 

2 



5 

5 

5 

5 

5 

5 

5.2 

5.2 



.363364 
363940 
364515 
365090 
365664 
366237 
366810 
367382 
367953 
368524 



.369663 
370232 
370799 
371367 
371933 
372499 
373064 
373629 
374193 
374756 

.375319 
375881 
376442 
377003 
377563 
378122 
378681 
379239 
379797 
380354 

.380910 
381466 
382020 
382576 
383129 
383682 
384234 
384786 
385337 



.386438 
386987 
387536 
388084 
388631 
389178 
389724 
390270 
390815 
391360 

.391903 
392447 
392989 
393531 
394073 
394614 
395154 
395694 
396233 
396771 



Cotang. 



10 



N.sine N. cos. 



10 



10 



10 



10 



10 



636636 
636060 
635485 
634910 
634336 
633763 
633190 
632618 
632047 
631476 
630906 
630337 
629768 
629201 
628633 
628067 
627501 
626936 
626371 
625807 
625244 
624681 
624119 
623558 
622997 
622437 
621878 
621319 
620761 
620203 
619646 
619090 
618534 
617980 
617425 
616871 
616318 
615766 
615214 
614663 
614112 
613562 
613013 
612464 
611916 
611369 
610822 
610276 
609730 
609185 
608640 
608097 
607553 
607011 
606469 
605927 
605386 
604846 
604306 
603767 
603229 



i! 22495 
N22523 
1 122552 
| 22580 
I 22608 
1 122637 
,22665 
| j 22693 
ii 22722 



22750 



1 22778 
i 122807 
i J 22835 



97437 
97430 
97424 
97417 
97411 
97404 
97398 
97391 
97384 
97378 
97371 
97365 
97358 



2286397351 

22892197345 



JI22920 
' 22948 
22977 
23005 
23033 
23062 
23090 
23118 
23146 
23176 
23203 
23231 
23260 
23288 
23316 
23345 
23373 
23401 
23429 
23458 
23486 
23514 
23542 
23571 
23599 
j 1 23627 
23656 
23684 
23712 
23740 
23769 
23797 
23825 
23853 
23882 
23910 
23938 
23966 
23995 
24023 
24051 
24079 
24108 
24136 
24164 
24192 



973b8 
97331 
97325 
97318 
97311 
97304 
97298 
97291 
97284 
97278 
97271 
97264 
97257 
97251 
97244 
97237 
97230 
97223 
97217 
97210 
97203 
97196 
97189 23 



Tang. 



97182 
97176 
9716SJ 
97162 
97155 
97148 
97141 
97134 
97127 
97120 
97113 
9710o 
97100 
97093 
97086 
97079 
97072 
97065 
97058 
97051 
97044 
97037 
97030 
N. cos N.sine. 



TABLE II. 



Log. Sines and Tangents. (14°) Natural Sines. 



35 




1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



(.383675 
384182 
384687 
385192 
385697 
386201 
386704 
387207 
387709 
388210 
388711 

1.389211 
389711 
390210 
390708 
391206 
391703 
392199 
392695 
393191 
393685 

1.394179 
394673 
395166 
395658 
396150 
396641 
397132 
397621 
398111 
398600 

.399088 
399575 
400062 
400549 
401035 
401520 
402005 
402489 
402972 
403455 

.403938 
404420 
404901 
405382 
405862 
406341 
406820 
407299 
407777 
408254 

.408731 
409207 
409682 
410157 
410632 
411106 
411579 
412052 
412524 
412996 
Cosine. 



D. lo' 



84.4 
84.3 
84.2 
84.1 
81.0 
83.9 
83.8 



83 

S3 

83 

83 

83 

83 

83 

83 

82.8 

82.7 

82.6 

82.6 

82.4 

82.3 

82.2 

82.1 

82.0 

81.9 

81.8 

81.7 

81.7 

81.6 

81.5 

81.4 

81.3 

81.2 

81.1 

81.0 

80.9 

80.8 

80.7 

80.6 

80.5 

80.4 

80.3 

80.2 

80.1 

80.0 

79.9 

79 

79 

79 



79.0 



78.8 
78.7 
78.6 



Cosine. 



.986904 
986873 
986841 
986809 
986778 
986746 
986714 
986683 
986651 
986619 
986587 

.986555 
986523 
986491 
986459 
986427 
986395 
986363 
986331 
986299 
986266 

.986234 
986202 
986169 
986137 
986104 
986072 
986039 
986007 
985974 
985942 

,985909 
985876 
985843 
985811 
985778 
985745 
985712 
985679 
985646 
985613 

.985580 
985547 
985514 
985480 
985447 
985414 
985380 
985347 
985314 
985280 

.985247 
985213 
985180 
985146 
985113 
985079 
985045 
985011 
984978 
984944 
Sine. 



D. 10' 



5.2 
5.3 
5.3 
5.3 
5.3 
5.3 
5.3 
5.3 



5.3 
5.3 
5.3 

5.4 
6.4 
5.4 
5.4 



5.4 
5.4 
5.4 
5.4 
5.4 
5.4 
5.5 
5.5 
5.5 
5.6 
5.5 
5.6 



6.6 
5.5 
6.5 
5.5 
5.6 
5.6 
6.6 
5.6 
5.6 



5.6 
6.6 
6.6 



Tang. 



396771 
397309 
397846 
398383 
398919 
399455 
399990 
400524 
401058 
401591 
402124 

9.402656 
403187 
403718 
404249 
404778 
405308 
405836 
406364 
406892 
407419 

9. 407945 
408471 
408997 
409521 
410045 
410569 
411092 
411615 
412137 
412658 

». 413179 
413699 
414219 
414738 
415257 
415775 
416293 
416810 
417326 
417842 

9.418358 
418873 
419387 
419901 
420416 
420927 
421440 
421952 
422463 
422974 
. 423484 
423993 
424503 
425011 
425519 
426027 
426534 
427041 
427647 
428052 
Cotang. 



D. 10" 



89.6 
89.6 
89.5 
89.4 
89.3 
89.2 
89.1 
89.0 
88.9 
88.8 
88.7 
88.6 
88.5 
88.4 
88.3 
88.2 
88.1 
88.0 
87.9 
87.8 
87.7 
87.6 
87.5 
87.4 
87.4 
87.3 
87.2 
87.1 
87.0 
86.9 
86.8 
86.7 
86.6 
86.5 
86.4 
86.4 
86.3 
86.2 
86.1 
86.0 
85.9 
85.8 
85.7 
85.6 
85.5 
86.5 
85.4 
85.3 
86.2 
85.1 
85.0 
84.9 
84.8 
84.8 
84.7 
84.6 
84.5 
84.4 
84.3 
84.3 



Cotang. 



10.603229 
602691 
602154 
601617 
-601081 
600545 
600010 
599476 
598942 
598409 
597876 

10.597344 
596813 
596282 
595751 
595222 
594692 
594164 
593636 
593108 
592581 

10.592055 
591529 
591003 
590479 
689955 
589431 



N. sine. N. cos 



588385 
587863 
587342 

10.586821 
586301 
685781 
685262 
584743 
684225 
583707 
683190 
582674 
582158 

10.581642 
681127 
580613 
580099 
579585 
679073 
678560 
578048 
577637 
577026 

10.576616 
676007 
675497 
574989 
574481 
573973 
673466 
572959 
572453 
571948 
Tan? 



24192 

24220 

24249 

24277 

24305 

24333 

24362 

24390 

24418 

24446 

24474 

24503 

24531 

24559 

24587 

24615 

24644 

24672 

24700 

24728 

24756 

24784 

24813 

24841 

24869 

24897 

24925 

24954 

24982 

25010 

25038 

25066 

25094 

25122 

25151 

25179 

25207 

25235 

25263 

25291 

25320 

25348 

25376 

25404 

25432 

25460 

25488 
125516 
j 25545 

25573 

25601 

25629 

25657 
| 25685 
I 25713 
(25741 

25766 
i25798 
125826 
i 25854 
| 25882 

N. cos, N.sim 



97030 
97023 
97015 
97008 
97001 
96994 
96987 
96980 
96973 
96966 
96959 
96952 
96945 
96937 
96930 
96923 
96916 
96909 
96902 
96894 
96887 
96880 
96873 
96866 
96858 
96851 
96844 
96837 
96829 
96822 
96815 
96807 
96800 
96793 
96786 
96778 
96771 
96764 
96756 
96749 
96742 
96734 
96727 
96719 
96712 
96705 
96697 
96690 
96682 
96676 
96667 
96660 
96653 
96645 
96638 
96630 
96623 
96615 
)6608 
96600 
96593 



75 Degrees. 



35 



Log. Sines and Tangents. (15°) Natural Sines. 



Sine. 



10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
38 
87 



40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



.412998 
413467 
413938 
414408 
414878 
415347 
415815 
416283 
416751 
417217 
417684 

.418160 
418615 
419079 
419544 
420007 
420470 
420933 
421395 
421857 
422318 

.422778 
423238 
423697 
424156 
424615 
425073 
425530 
425987 
426443 
426899 

'.427354 
427809 
428263 
428717 
429170 
429623 
430075 
430527 
430978 
431429 

.431879 
432329 
432778 
433226 
433675 
434122 
434569 
435016 
435462 
435908 

.436353 
436798 
437242 
437686 
438129 
438572 
439014 
439456 
439897 
440338 

Cosine. 



D. 10"| Cosine. 



78.5 
78.4 
78.3 
78.3 
78.2 
78.1 
78.0 
77.9 
77.8 
77.7 
77.6 
77.5 
77.4 
77.3 
77.3 
77.2 
77.1 
77.0 
76.9 
76.8 
76.7 
76.7 
76.6 
76.5 
76.4 
76.3 
76.2 
76.1 
76.0 
76.0 
75.9 
75.8 
75.7 
75.6 
75.5 
75.4 
75.3 
75.2 
75.2 
75.1 
75.0 
74.9 
74.9 
74.8 
74.7 
74 6 
74.5 
74.4 
74.4 
74.3 
74.2 
74.1 
74.0 
74.0 
73.9 
73.8 
73.7 
73.6 
73.6 
73.5 



984944 
984910 
984876 
984842 
984808 
984774 
984740 
984706 
984672 
984637 
984603 
984569 
984535 
984500 
984466 
984432 
984397 



984328 
984294 
984259 
.984224 
984190 
984155 
984120 
984085 
984050 
984015 
983981 
983946 
983911 
.983875 
983840 



9837^0 
983735 
983700 



983629 



983558 
.983523 
983487 
983452 
983416 
983381 
983345 
983309 
983273 
983238 
983202 
.983166 
983130 
983094 
983058 
983022 



982950 
982914 

982878 
982842 



Sine. 



D. 10"| Tang. 



5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.8 
6.8 
5.8 
5.8 
6.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
6.8 
5.9 
6.9 
6.9 
5.9 
5.9 
5.9 
5.9 
5.9 
5.9 
5.9 
5.9 
5.9 
6.9 
5.9 
6.9 
5.9 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 



9.428052 
428557 
429062 
429566 
430070 
430573 
431075 
431577 
432079 
432580 
433080 

9.433580 
434080 
434579 
435078 
435576 
436073 
436570 
437067 
437563 
438059 

9.438554 
439048 
439643 
440036 
440529 
441022 
441514 
442006 
442497 
442988 

1.443479 
443968 
444458 
444947 
445435 
446923 
446411 
446898 
447384 
447870 

1.448356 
448841 
449326 
449810 
450294 
450777 
451260 
461743 
462226 
452706 

1.453187 
453668 
454148 
454628 
455107 
465586 
456064 
456642 
457019 
457496 
Cotang. 



D. 10" 

84.2 
84.1 
84.0 
83.9 
83.8 
83.8 
83.7 
83.6 
83.6 
83.4 
83.3 
83.2 
83.2 
83.1 
83.0 
82.9 
82.8 
82.8 
82.7 
82.6 
82.5 
82.4 
82.3 
82.3 
82.2 
82.1 
82.0 
81.9 
81.9 
81.8 
81.7 
81.6 
81.6 
81.5 
81.4 
81.3 
81.2 
81.2 
81.1 
81.0 
80.9 
80.9 
80.8 
80.7 
80.6 
80.6 
80.5 
80.4 
80.3 
80.2 
80.2 
80.1 
80.0 
79.9 
79.9 
79.8 
79.7 
79.6 
79.6 
79.5 



Cotang. N. sine 

10.571948 

571443 

570938 

570434 

569930 

669427 

668925 

568423 

667921 

667420 

566920 
10.566420 

565920 

565421 

564922 

664424 

663927 

663430 

662933 

662437 

661941 
10.561446 

660952 

560457 

559964 

559471 

658978 

558486 

557994 

557503 

657012 
10.556521 

556032 

655542 

655053 

554665 

654077 

653589 

653102 

552616 

552130 
10.551644 

551159 

650674 

650190 

649706 

649223 

548740 

648257 

547775 

547294 
10.546813 

546332 

545852 

646372 

544893 

644414 

543936 

543458 

542981 

542504 



Tang. 



N. cos-. 

96593 
96585 
96578 
96570 
96562 
96555 
96547 
96540 
96532 
96524 
96517 
96509 
96502 
96494 
96486 
96479 
96471 
96463 
96456 
96448 



25882 
25910 
25938 
25966 
25994 
26022 
26050 
26079 
26107 
26135 
26163 
26191 
26219 
26247 
26275 
26303 
26331 
26359 
26387 
26415 
26443196440 



26471 
26500 
26528 
26556 
26584 
26612 
26640 
26668 
26696 
26724 
26752 
26780 
26808 
26836 
26864 
26892 
26920 
26948 
26976 
27004 
27032 
27060 
27088 
27116 
27144 
27172 
27200 
27228 
27256 
27284 
27312 
27340 
27368 
27396 
27424 
27452 
27480 
27508 
27536 
27564 



96433 
96425 
96417 
96410 
96402 
96394 
96386 
96379 
96371 
96363 
96355 
96347 
96340 
96332 
96324 
96316 
96308 
96301 
96293 
96285 
96277 
96269 
96261 
96253 
96246 
96238 
96230 
96222 
96214 
96206 
96198 
96190 
96182 
96174 
96166 
96158 
96150 
96142 
96134 
96126 



N. cos. J N. sine. ' 



74 Degrees. 



TABLE II. 



Log. Sines and Tangents. (16°) Natural Sines. 



37 



Sine. 

.410338 
440778 
441218 
441658 
442096 
442535 
442973 
443410 
443847 
444284 
444720 

.445155 
445590 
446025 
446459 
446893 
447326 
447759 
448191 
448623 
449054 

.449485 
449916 
450345 
450775 
451204 
451632 
452060 
452488 
452915 
453342 

.453768 
454194 
454619 
455044 
455469 
455893 
456316 
456739 
457162 
457584 

.458006 
458427 
458848 
459268 

'459688 
460108 
460527 
460946 
461364 
461782 

.462199 
462616 
463032 
463448 
463864 
464279 
464694 
465108 
465522 
465935 



Cosine. 



D. 10' 



73.4 
73.3 
73.2 
73-1 
73.1 
73.0 
72.9 
72.8 
72.7 
72.7 
72.6 
72.5 
72.4 
72.3 
72.3 
72.2 
72.1 
72.0 
72.0 
71.9 
71.8 
71.7 
71.6 
71.6 
71.5 
71.4 
71.3 
71.3 
71.2 
71.1 
71.0 
71.0 
70.9 
70.8 



70 

70 

70 

70 

70 

70 

70 

70 

70 

70.1 

70.0 

69.9 

69.8 

69.8 

69.7 

69.6 

69.5 

69.5 

69.4 

69.3 

69.3 

69.2 

69.1 

69.0 

69.0 

68.9 



Cosine. 



982842 
982805 
982769 
982733 
982696 
982660 
982624 
982587 
982551 
982514 
982477 
982441 
982404 
982367 
982331 
982294 
982257 
982220 
982183 
982146 
982109 
9.982072 
982035 
981998 
981961 
981924 
981886 
981849 
981812 
981774 
981737 
981699 
981662 
981625 
981587 
981549 
981512 
981474 
981436 
981399 
981361 
981323 
981285 
981247 
981209 
981171 
981133 
981095 
981057 
981019 
980y81 
980942 
980904 
980866 
980827 
980789 
980750 
980712 
980673 
980635 
980596 



Sine. 



D. 10' 



6.0 
6.0 
6.1 
6.1 
6.1 
6.1 



6 
6 
6 
6 
6 
6 
6 

6.1 
6.1 
6.1 
6.1 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.4 
6.4 
6.4 
4 
4 
4 
4 
4 
4 
4 
4 
6.4 
6.4 
6.4 



Tang. 



9.457496 
457973 
458449 
458925 
459400 
459875 
460349 
460823 
461297 
461770 
462242 

9 462714 
463186 
463658 
464129 
464599 
465069 
465539 
466008 
466476 
466945 

9 467413 
467880 
468347 
468814 
469280 
469746 
470211 
470676 
471141 
471605 

9 472068 
'472532 
472995 
473457 
473919 
474381 
474842 
475303 
475763 
476223 

9 476683 
'477142 
477601 
478059 
478517 
478975 
479432 
479889 
480345 
480801 

9 481257 
481712 
482167 
482621 
483075 
483529 
483982 
484435 
484887 
486339 



Cotang. 



D. 10" 



79.4 
79.3 
79.3 
79.2 
79.1 
79.0 
79.0 
78.9 
78.8 
78.9 
78.7 
78-6 
78.6 
78.5 
78.4 
78.3 
78.3 
78.2 
78.1 
78.0 
78.0 
77- 9 
77-8 
77-8 
77.7 
77.6 
77-5 
77-5 
77-4 
77-3 
77.3 
77.2 
77-1 
77.1 
77-0 
76.9 
76.9 
76.8 
76.7 
76.7 
76.6 
76.5 
76.5 
76.4 
76.3 
76.3 
76.2 
76.1 
76.1 
76.0 
75.9 
75.9 
75.8 
75.7 
75.7 
75.6 
75.5 
75.6 
75.4 
75.3 



Cotang. 



10.542504 
642027 
541551 
641076 
540600 
640126 
539651 
639177 
538703 
638230 
637758 

10.637286 
636814 
536342 
536871 
535401 
634931 
634461 
633992 
633524 
633055 

10.532587 
532120 
531653 
531186 
530720 
530254 
629789 
629324 
628859 
528395 

10.527932 
527468 
527005 
626543 
526081 
625619 
626158 
524697 
524237 
523777 

10.623317 
522858 
622399 
521941 
521483 
521025 
620568 
620111 
519655 
619199 

10.518743 
518288 
617833 
517379 
516925 
516471 
516018 
515565 
616113 
514661 



Tang. 



N. sine. N. cos 



27564 
27592 
27620 
27648 
27676 
27704 
27731 
27769 
27787 
27815 
27843 
27871 
27899 
27927 
27955 
27983 
28011 
28039 
28067 
28096 
28123 
28150 
28178 
28206 
28234 
28262 
28290 
28318 
28346 
28374 
28402 
28429 
28467 
28486 
28513 
28641 
28569 
28597 
28626 
28652 
28680 
28708 
28736 
28764 
28792 
28820 
28847 
28876 
28903 
28931 
28959 
28987 
29015 
29042 
29070 
29098 
29126 
29154 
29182 
29209 
29247 



N. cos. N.sine 



96126 
96118 
96110 
96102 
96094 
96086 
96078 
96070 
96062 
96054 
96046 
96037 
96029 
96021 
96013 
96005 
y5997 
95989 
95981 
95972 
95964 
95956 
95948 
95940 
95931 
95923 
95915 
95907 



96890 



95874 
96865 
95867 
95849 
95841 
95832 
95824 
95816 
H5807 
95799 
95791 
95782 
95774 
95766 
95767 
95749 
96740 
95732 
95724 
95715 
95707 



95690 
95681 
95673 
95664 
95656 
95647 
95639 
95630 



73 Degrees. 



38 



Log. Sines and Tangents. (17°) Natural Sines. TABLE II. 



Sine. 



9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



9.465935 
466348 
466761 
467173 
467585 
467996 
468407 
468817 
469227 
469637 
470046 
9.470455 
470863 
471271 
471679 
472086 
472492 
472898 
473304 
47S710 
474115 
9.474519 
474923 
475327 
475730 
476133 
476536 
476938 
477340 
477741 
478142 
9.478542 
478942 
479342 
479741 
480140 
480539 
480937 
481334 
481731 
482128 
482525 
482921 
483316 
483712 
484107 
484501 
484895 
485289 
485682 
486075 
9.486467 
486860 
487^51 
487643 
488034 
488424 
488814 
489204 
489593 
489982 



Cosine. 



68.8 

68.8 

68.7 

68.6 

68.5 

68.5 

68.4 

68.3 

68.3 

68.2 

68.1 

68.0 

68.0 

67.9 

67.8 

67.8 

67.7 

67.6 

67.6 

67.5 

67.4 

67.4 

67.3 

67.2 

67.2 

67.1 

67.0 

66.9 

66.9 

66.8 

66.7 

66.7 

66.6 

66.5 

66.5 

66.4 

66.3 



Cosine. 

1.980596 
980558 
980519 
980480 
980442 



980364 
980325 
980286 
980247 



66.1 
66.0 
65.9 
65.9 
65.8 
65.7 
65.7 
65.6 
65.5 
65.5 
65.4 
66.3 
65.3 
65.2 
65.1 
65.1 
65.0 
65.0 
64.9 
64.8 



'.980169 
980130 
980091 
980052 
980012 
979973 
979934 
979895 
979855 
979816 
.979776 
979737 
979697 
979658 
979618 
979579 
979539 
979499 
979459 
979420 
.979380 
979340 
979300 
979260 
979220 
979180 
979140 
979100 
979059 
979019 
.978979 
978939 
978898 
978858 
978817 
978777 
978736 
978696 
978655 
978615 
.978574 
978533 
978493 
978452 
978411 
978370 
978329 
978288 
978247 
978206 



Sine. 



6 

6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 



7 

7 

7 

7 

7 

7 

7 

7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 



Tang. 

9.485339 

485791 
486242 
486693 
487143 
487593 
488043 
488492 
488941 
489390 



D. 10" 



.490286 
490733 
491180 
491627 
492073 
492519 
492965 
493410 
493854 
494299 
.494743 
495186 
495630 
496073 
496515 
496957 
497399 
497841 
468282 
498722 
499163 
499603 
500042 
500481 
500920 
501359 
501797 
502235 
502672 
503109 
503546 
503982 
504418 
504854 
505289 
505724 
506159 
506593 
507027 
507460 
9.507893 
508326 
508759 
509191 
509622 
510054 
510485 
510916 
511346 
511776 



Cotang. 



75.3 

75.2 

75.1 

75.1 

75.0 

74.9 

74.9 

74.8 

74~. 7 

74.7 

74.6 

74.6 

74.5 

74.4 

74.4 

74.3 

74.3 

74.2 

74.1 

74.0 

74.0 

74.0 

73.9 

73.8 

73.7 

73.7 

73.6 

73.6 

73.5 

73.4 

73.4 

73.3 

73.3 

73.2 

73.1 

73.1 

73.0 

73.0 

72.9 

72.8 

72.8 

72.7 

72.7 

72.6 



Cotang. |N. sine. N. cos 



72.5 
72.6 
72.4 
72.4 
72.3 
72.2 
72.2 
72.1 
72.1 
72.0 
71.9 
71.9 
71.8 
71.8 
71.7 
71.6 



10.614661 
514209 
513758 
513307 
512857 
512407 
511957 
511508 
511059 
510610 
510162 
10.509714 
509267 
508820 
508373 
607927 
507481 
507035 
506590 
506146 
505701 
10.505257 
504814 
604370 
503927 
503485 
503043 
502601 
502159 
501718 
601278 
10.500837 
500397 
499958 
499519 
499080 
498641 
498203 
497765 
497328 
496891 
10.496454 
496018 
495582 
495146 
494711 
494276 
493841 
493407 
492973 
492540 
10.492107 
491674 
491241 
490809 
490378 
489946 
489515 
489084 
488654 
488224 



1 29237 
29265 
29293 
29321 
29348 
29376 
29404 
29432 
29460 
2948 
29515 
29543 
29571 
29599 
29626 
29654 
29682 
29710 
29737 
29765 
29793 
29821 



2984995441 



29876 

29904 

29932 

29960 

29987 

30015 

30043 

30071 

30098 

30126 

30154 

30182 

30209 

30237 

30265 

30292 

30320 

30348 

30376 

30403 

30431 

30459 

30486 

30514 

30542 

30570 

3059 

30625 

30653 

30680 



95630 
95622 
95613 
95605 
95596 
95588 
95579 
95671 
95562 
95554 
'95545 
95536 
95528 
95519 
95511 
95502 
95493 
95485 
95476 
95467 
95459 
95450 



30763 
30791 
30819 
30846 
30874 
30902 



Tang. || N. cos 



95433 
95424 
95415 
95407 
95398 
95389 
95380 
95372 
95363 
95354 
95346 
95337 
95328 
95319 
95310 
95301 
95293 
95284 
95275 
95266 
95257 
95248 
95240 
95231 
95222 
95213 
95204 
95195 
95186 
95177 



30708 95168 
30736 95159 



95150 
95142 
95133 
95124 
95115 
95106 



N.sinc 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

U 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 




TABLE II. 



Log. Sines and Tangents. (18°) Natural Sines. 



39 



Cosine. 

.978206 
978165 
978124 
978083 
978042 
978001 
977959 
977918 
977877 
977835 
977794 

.977752 
977711 
977669 
977628 
977586 
977544 
977503 
977461 
977419 
977377 

.977335 
977293 
977251 
977209 
977167 
977125 
977083 
977041 
976999 
976957 

.976914 
976872 
976830 
976787 
976745 
976702 
976660 
976617 
976574 
976532 

.976489 
976446 
976404 
976361 
976318 
976275 
976232 
976189 
976146 
976103 

>. 976060 
976017 
976974 
975930 
975887 
975844 
975800 
975757 
975714 
975670 



D. 10" 



Tang. 



D. 10" 



Cotang. i N. sine. N. cos. 



Sine. 

489982 
490371 
490759 
491147 
491535 
491922 
492308 
492695 
493081 
493466 
493851 

9.494236 
494621 
495005 
495388 
495772 
496154 
496537 
496919 
497301 
497682 

9 498064 
498444 
498825 
499204 
499584 
499963 
500342 
500721 
501099 
501476 
501854 
502231 
502607 
502984 
503360 
503735 
504110 
504485 
504860 
505234 

9.505608 
505981 
506354 
506727 
507099 
507471 
507843 
508214 
508585 
508956 
.509326 
509696 
510065 
510434 
510803 
511172 
511540 
511907 
512275 
512642 
Cosine. 



D. 10' 



64.8 

64.8 

64.7 

64.6 

64.6 

64.5 

64.4 

64.4 

64.3 

64.2 

64.2 

64.1 

64.1 

64.0 

63.9 

63.9 

63.8 

63.7 

63.7 

63.6 

63.6 

63.5 

63 

63 

63 

63 

63 

63 

63 

63.0 

62.9 

62.9 

62.8 

62.8 

62.7 

62.6 

62.6 

62.5 

62.5 

62.4 

62.3 

62.3 

62.2 

62.2 

62.1 

62.0 

62.0 

61.9 

61.9 

61.8 

61.8 

61.7 

61.6 



Sine. 



;.8 



6.9 
6.9 
6.9 
6.9 
6.9 
6.9 



6.9 
6.9 
6.9 
6.9 



7.0 
7.0 



7.1 
7.1 
7.1 

7.1 

7.2 



511776 
512206 
512635 
513064 
513493 
513921 
514349 
514777 
515204 
515631 
516057 
516484 
516910 
517335 
517761 
518185 
518610 
519034 
519458 
519882 
620305 
520728 
521151 
621573 
521995 
522417 
522838 
523259 
523680 
624100 
524520 
524939 
625359 
525778 
526197 
526615 
527033 
627451 
527868 
528285 
628702 
9.529119 
529535 
529950 
530366 
530781 
531196 
531611 
532025 
532439 
532853 
.533266 
633679 
534092 
534504 
534916 
535328 
535739 
636150 
536561 
536972 
Cotang. 



71.6 

71.6 

71.5 

71.4 

71 

71 

71 

71 

71 

71 

71 

71.0 

70.9 

70.9 

70.8 

70.8 

70 

70 

70 

70 

70 

70 

70 

70.3 

70.3 

70.2 

70.2 

70.1 

70.1 

70.0 

69.9 

69.9 



69.7 
69.7 
69.6 
69.6 
69.5 



5 
4 
3 
3 
3 
2 
69.1 
69.1 
69.0 
69.0 
68 9 
68.9 
68.8 
68.8 
68.7 
68.7 
68 6 
68.6 
68.5 
68.5 
68.4 



10.488224 
487794 
487365 
486936 
486507 
486079 
485651 
485223 
484796 
484369 
483943 

10.483516 
483090 
482665 
482239 
481815 
481390 
480966 
480542 
480118 
479695 

10.479272 
478849 
478427 
478005 
477583 
477162 
476741 
476320 
475900 
475480 

10-475061 
474641 
474222 
473803 
473385 
472967 
472549 
472132 
471715 
471298 

10.470881 
470465 
470050 
469634 
469219 
468804 
468389 
467976 
467561 
467147 

10.466734 
466321 
465908 
465496 
465084 
464672 
464261 
463850 
463439 
463028 
Tang. 



30902 
30929 
30957 
30985 
31012 
31040 
31068 
31095 
31123 
31151 
31178 
31206 
31233 
31261 
31289 
31316 
31344 
31372 
31399 
31427 
31454 
31482 
31510 
31537 
31565 
31593 



95106 
95097 
95088 
95079 
95070 
95061 
95052 
95043 
95033 
95024 
95015 
95006 
94997 
94988 
94979 
94970 
94961 
94952 
94943 
94933 
94924 
94915 
94906 
94897 
94888 
94878 



3162094869 
31648194860 



31675 
31703 
31730 



94851 
94842 
94832 



31758 94823 



31786 
31813 
31841 
31868 
31896 
31923 
31951 
31979 
32006 



94814 
94805 
94795 
94786 
94777 
94768 
94758 
94749 
94740 



32034 94730 
32061 94721 
32089 94712 
32116 94702 
32144 94693 
3217194684 
3219994674 
3222794665 
3226094656 
32282 94646 
32309 94637 



32337 
32364 
32392 
32419 
32447 
32474 
32502 
32529 
32557 



94627 
94618 
94609 
94599 
94590 
94580 
94571 
94561 
94552 
N. cos. N.sine. 



71 Degrees. 



40 



Log. Sines and Tangents. (19°) Natural Sines. TABLE II. 





1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 



Sine. 



9.512642 
613009 
513375 
513741 
514107 
514472 
514837 
515202 
515566 
515930 
516294 
516657 
517020 
517382 
517745 
518107 
518468 
518829 
619190 
519551 
619911 
9.620271 
620631 
620990 
621349 
521707 
522066 
522424 
622781 
623138 
623495 
9.623852 
524208 
624564 
524920 
526275 
625 630 
525984 
626339 
626693 
627046 
527400 
627753 
528105 
628458 
528810 
529161 
629513 
629864 
530216 
630565 
530915 
631266 
631614 
531963 
632312 
632661 
533009 
533357 
533704 
634052 



D. lu' 



Cosine. 



61.2 

61.1 

61.1 

61.0 

60.9 

60.9 

60.8 

60.8 

60.7 

60.7 

60.6 

60.5 

60.6 

60.4 

60.4 

60.3 

60-3 

60-2 

60.1 

60.1 

60.0 

60.0 

59.9 

59.9 

59.8 

59.8 

59.7 

59.6 

59.6 

59.5 

59.5 

59.4 

59.4 

59.3 

59.3 

59-2 

59.1 

59.1 

59.0 

59.0 

58-9 

58.9 

58.8 

58.8 

58.7 

68. 7 

58-6 

58-6 

58.5 

68.5 

68.4 

58-4 

68.3 

58-2 

68-2 

68-1 

58.1 

68.0 

58.0 

67.9 



Cosine. 

9.975670 
975627 
975583 
975539 
975496 
975452 
976408 
975365 
975321 
975277 
975233 
9.976189 
975145 
975101 
975057 
975013 
974969 
974925 
974880 
974836 
974792 
974748 
974703 
974659 
974614 
974570 
974525 
974481 
974436 
974391 
974347 
9.974302 
974257 
974212 
974167 
974122 
974077 
974032 
973987 
973942 
973897 
973852 
973807 
973761 
973716 
973671 
973625 
973580 
973636 
973489 
973444 
9.973398 
973352 
973307 
973261 
973215 
973169 
973124 
973078 
973032 
972986 



D. lu 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.6 

7.6 

7.5 

7.5 

7.5 

7.5 

7.5 

7.5 

7.6 

7.5 

7.6 

7.5 

7.5 

7.5 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.7 



Sine. | 



Tam* ~ 

9.636972 
637382 
637792 
538202 
638611 
639020 
639429 
539837 
540245 
540653 
641061 
9.541468 
641875 
542281 
642688 
643094 
543499 
543905 
644310 
644716 
645119 
9.545624 
545928 
646331 
646736 
547138 
547540 
547943 
648345 
648747 
649149 
9.549550 
649951 
650362 
650752 
661152 
561552 
551952 
652351 
562760 
653149 
9.553548 
653946 
554344 
654741 
656139 
655536 
555933 
656329 
666726 
557121 
557617 
557913 
658308 
658702 
559097 
659491 
659885 
560279 
660673 
561066 



1). lu 



Cotang. 



68.4 

68.3 

68.3 

68.2 

68.2 

68.1 

68.1 

68.0 

68.0 

67:9 

67.9 

67.8 

67.8 

67.7 

67.7 

67.6 

67.6 

67.6 

67.5 

67.4 

67.4 

67.3 

67.3 

67.2 

67.2 

67.1 

67.1 

67.0 

67.0 

66.9 

66.9 

66.8 

66.8 

66.7 

66.7 

66.6 

66.6 

66.5 

66.5 

66.6 

66.4 

66.4 

66.3 

66.3 

66.2 

66.2 

66.1 

66.1 

66.0 

66.0 

66.9 

65.9 

66.9 

66.8 

65.8 

65.7 

65.7 

65.6 

65.6 

65.5 



Cotang. 



10.463028 
462618 
462208 
461798 
461389 
460980 
460571 
460163 
459755 
459347 
458939 
10.458532 
468125 
457719 
457312 
456906 
456501 
466095 
455690 
455285 
454881 
10.454476 
454072 
453669 
453265 
452862 
452460 
452057 
451655 
451253 
450851 
10.450450 
450049 
449648 
449248 
448848 
448448 
448048 
447649 
447250 
446851 
10.446452 
446054 
445656 
445259 
444861 
444464 
444067 
443671 
443275 
442879 
10.442483 
442087 
441692 
441298 
440903 
440509 
440116 
439721 
439327 
438934 



N. sine. N. cos 



32557 

32584 

32612 

32639 

32667 

32694 

32722 

32749 

32777 

32804 

32832 

32859 

32887 

32914 

32942 

32969 

32997 

33024 

33051 

33079 

33106 

33134 

33161 

33189 

33216 

33244 

33271 

33298 

33326 

33353 

33381 

33408 

33436 

33463 

33490 

33518 

33545 

33673 



94552 

94542 

94533 

94523 

94514 

94504 

94495 

94485 

94476 

94466 

94457 

94447 

94438 

94428 

94418 

94409 

94399 

94390 

94380 

94370 

94361 

94351 

94342 

94332 

94322 

94313 

94303 

94293 

94284 

94274 

94264 

94254 

94245 

94235 

94225 

94216 

94208 

94196 



3360094186 



Tang. 



33627 
33655 
33682 
33710 
33737 
33764 
83792 
33819 
33846 
33874 
33901 
33929 
33956 
33983 
34011 
34038 
34065 
34093 
34120 



34147 93989 



34175 



94176 
94167 
94167 
94147 
94137 
94127 
94118 
94108 
94098 
94088 
94078 
94068 
94058 
94049 
94039 
94029 
94019 
94009 
93999 



93979 



34202 93969 



N. cos. N.sine. 



60 
59 
68 
57 
56 
65 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
3S 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 




70 Degrees. 



Log. Sines and Tangents. 



D ) Natural Sine.". 



41 



Sine. 

). 534052 
534399 
534745 
535092 
535438 
636783 
536129 
536474 
636818 
537163 
537507 

9.537851 
538194 
538538 
638880 
539223 
539565 
539907 
540249 
540590 
640931 
.541272 
541613 
641953 
542293 
542632 
542971 
543310 
543649 
643987 
644325 

9.644663 
545000 
645338 
645674 
546011 
546347 
646683 
647019 
547354 
647689 

9.648024 
648359 
648693 
549027 
549360 
649693 
650026 
550359 
650692 
551024 

9.651366 
551687 
552018 
552349 
552680 
553010 
653341 
553670 
654000 
554329 



Cosine. 



\). 10' 



57.8 



57 

57 

67 

57 

57 

57 

57 

57 

57 

57 

67 

57 

67.1 

57.1 

57.0 

57.0 

56.9 

66.9 

56 

56 

56 

56 

66 

66 

56 

56 



56.4 



66 

56 

56 

56 

56 

56 

66 

56.0 

56.0 

66.9 

65.9 

55.8 

55.8 

65.7 

55.7 

55.6 

65.6 

55.6 

65.6 

55.4 

65.4 

55.3 

55.3 

66.2 

65.2 

55.2 

65.1 

65.1 

55.0 

65.0 

64.9 

64.9 



Cosine. 

9.972986 
972940 
972894 
972848 
972802 
972755 
972709 
972663 
972617 
972570 
972524 

9.972478 
972431 
972385 
972338 
972291 
972245 
972198 
972161 
972105 
972058 

9.972011 
971964 
971917 
971870 
971823 
971776 
971729 
971682 
971635 
971588 

9.971540 
971493 
971446 
971398 
971351 
971303 
971256 
971208 
971161 
971113 

9.971066 
971018 
970970 
970922 
970874 
970827 
970779 
970731 
970683 
970635 

9.970586 
970538 
970490 
970442 
970394 
970346 
970297 
970249 
970200 
970152 



Sine. 



I). 10' 



7.7 
7.7 
7.7 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.1 
8.1 
8.1 
8.1 



[rang. 

9.661066 
561469 
561861 
662244 
662636 
663028 
663419 
563811 
664202 
564592 
664983 

9.665373 
665763 
666163 
666542 
666932 
567320 
567709 
668098 
668486 
668873 
569261 
569648 
670035 
670422 
670809 
671196 
671581 
671967 
672352 
572738 
573123 
573507 
673892 
674276 
674660 
675044 
675427 
675810 
576193 
676576 

9.576968 
577341 
577723 
578104 
578486 
578867 
579248 
679629 
580009 
680389 

9.580769 
681149 
681528 
681907 
682286 
582665 
583043 
683422 
683800 
684177 

Cotang. 
Degrees. 



I). 10' 



65.5 
65.4 
65.4 
65.3 
65.3 
65.3 
65.2 
65.2 
66.1 
65.1 
65.0 
66.0 
64.9 
64.9 
64.9 
64.8 
64.8 
64.7 
64.7 
64.6 
64.6 
64.5 
64.5 
64.5 
64.4 
64.4 
64.3 
64.3 
64.2 
64.2 
64.2 
64.1 
64.1 
64.0 
64.0 
63.9 
63.9 
63.9 
63.8 
63.8 
63.7 
63.7 
63.6 
63.6 
63.6 
63.6 
63.5 
63.4 
63.4 
63.4 
63.3 
63.3 
63.2 
63.2 
63.2 
63.1 
63.1 
63.0 
63.0 
62.9 



Uotang. 

10.438934 
438541 
438149 
437756 
437364 
436972 
436581 
436189 
435798 
435408 
435017 

10.434627 
434237 
433847 
433458 
433068 
432680 
432291 
431902 
431514 
431127 

10.430739 
430362 
429966 
429578 
429191 
428805 
428419 
428033 
427648 
427262 

10.426877 
426493 
426108 
425724 
425340 
424956 
424673 
424190 
423807 
423424 

10.423041 
422659 
422277 
421896 
421514 
421133 
420752 
420371 
419991 
419611 

10.419231 
418851 
418472 
418093 
417714 
417335 
416957 
416578 
416200 
415823 
Tang. 



34202 
34229 
34257 
34284 
34311 
34339 
34366 
34393 
34421 
34448 
34475 
34503 
34530 
34567 
34584 
34612 
34639 
34666 
34694 
34721 
34748 
34776 
34803 
34830 
34867 
34884 
34912 
34939 
34966 
34993 
35021 
35048 
35075 
35102 
35130 
35157 
35184 
35211 
35239 
35266 
35293 
35320 
36347 
35375 
35402 
35429 
35466 
35484 
35511 
35538 
36665 
36692 
35619 
35647 
35674 
36701 
35728 
36755 
35782 
35810 
35837 



93969 

93959 
93949 
93939 
93929 
93919 
9390!) 
98899 
93889 
98879 
98869 
93859 
93849 
93839 
93829 
93819 
93809 
93799 
93789 
93779 
93769 
93759 
93748 
93738 
93728 
93718 
93708 
93698 
98688 
93677 
93667 
93657 
93647 
93637 
98626 
93616 
93606 
93596 
93585 
93576 
93565 
93555 
93544 
93534 
93524 
93514 
93503 
93493 
93483 
93472 
93462 
93452 
93441 
93431 
93420 
93410 
93400 
93389 
93379 
93368 
93358 



N. cos. N.sine. ' 



42 



Log. Sines and Tangents. (21°) Natural Sines. 



TABLE II. 



10 
11 
12 

13 
14 

15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 



Sine. D. 10" Cosine, 



9.554329 
554658 
554987 
555315 
555643 
555971 
556299 
556626 
656953 
557280 
557606 

9.557932 
558258 
558583 
558909 
559234 
559558 
559883 
560207 
560531 
560855 

9.561178 
561501 
561824 
562146 
562468 
562790 
563112 
563433 
563755 
564075 
564396 
564716 
565036 
565356 
565676 
565995 
566314 
566632 
566951 
567269 
9.567587 
567904 
568222 
568539 
568856 
569172 
569488 
569804 
570120 
570435 

t. 570751 
571066 
571380 
571695 
672009 
672323 
572636 
572950 
573263 
573575 
Cosine. 



54.8 
54.8 
54.7 
54.7 
54.6 
54.6 
54.5 
54.5 
54.4 
54.4 
54.3 
54.3 
54.3 
54.2 
54.2 
54.1 
54.1 
54.0 
54.0 
53.9 
53.9 
53.8 
53.8 



53 

53 

53 

53 

53 

53 

53 

53 

53 

53 

63.3 

53.2 

53.2 

53.1 

53.1 

53.1 

53.0 

53.0 

52.9 

52.9 

52.8 

52.8 

52.8 

52.7 

52.7 

62.6 

52.6 

52.5 

52.5 

52.4 

52.4 

52.3 

52.3 

52.3 

52.2 

52.2 

52.1 



9.970152 
970103 
970055 
970006 
969957 
969909 
969860 
969811 
969762 
969714 
969665 
). 969616 
969567 
969518 
969469 
969420 
969370 
969321 
969272 
969223 
969173 
'.969124 
969075 
969025 
968976 
968926 
968877 
968827 
968777 
968728 
968678 
.968628 
968578 
968528 
968479 
968429 
968379 
968329 
968278 
968228 
968178 
.968128 
968078 
968027 
967977 
967927 
967876 
967826 
967775 
967725 
967674 
.967624 
967573 
967522 
967471 
967421 
967370 
967319 
967268 
967217 
967166 



Sine. 



8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 



8.2 
8.2 
8.2 
8.3 
8.3 



8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.5 
8.5 
8.5 
8.5 
8.5 
8.5 
8.5 



Tang. 



9.584177 
584555 
584932 
585309 
585686 
686062 
686439 
686816 
687190 
587566 
587941 
9.588316 
688691 
589066 
589440 
689814 
590188 
590562 
590935 
591308 
591681 
). 592054 
592426 
692798 
593170 
593542 
593914 
694285 
594656 
695027 
595398 
•.595768 
596138 
696508 
596878 
597247 
597616 
697985 
598354 
598722 
599091 
.699469 
599827 
600194 
600562 
600929 
601296 
601662 
602029 
602395 
602761 
.603127 
603493 
603858 
604223 
604588 
604953 
605317 
605682 
60)046 
606410 



D. 10' 



62.9 

62.9 

62.8 

62.8 

62.7 

62.7 

62.7 

62.6 

62,6 

62.5 

62.5 

62.5 

62 

62 

62 

62 

62 

62 

62 

62.2 

62.1 

62.1 

62.0 

62.0 

61.9 

61.9 

61.8 

61.8 

61.8 

61.7 

61.7 

61.7 

61.6 

61.6 

61.6 

61.5 

61.5 

61.5 

61.4 

61.4 

61.3 

61.3 

61.3 

61.2 

61.2 

61.1 

61.1 

61.1 

61.0 

61.0 

61.0 



Cotang. 



10.415823 
415445 
415068 
414691 
414314 
413938 
413561 
413185 
412810 
412434 
412059 

10.411684 
411309 
410934 
410560 
410186 
409812 
409438 
409065 
408692 
408319 

10.407946 
407574 
407202 
406829 
406458 



Cotang. 



60 

60 

60 

60 

60 

60 

60.7 
.7 
.6 



405715 
405344 
404973 
404602 
10.404232 



N.sinc 



403492 
403122 
402753 
402384 
402015 
401646 
401278 
400909 
10.400541 
400173 
399806 



399071 
398704 
398338 
397971 
397605 
397239 
10.396873 
396507 
396142 
395777 
395412 
395047 
394683 
394318 
393954 
393590 



35837 

35864 

35891 

■35918 

35945 

35973 

36000 

36027 

36054 

36081 

36108 

36135 

36162 

36190 

36217 

36244 

36271 

36298 

36325 

36352 

36379 

36406 

36434 

36461 

36488 

36515 

36542 

36569 

36596 

36623 

36650 

36677 

36704 

36731 

36758 

36785 

36812 

36839 

36867 



36921 
36948 
36975 
37002 
37029 
37056 
37083 
37110 
37137 
37164 
37191 
37218 
37245 
37272 
37299 
37326 
37353 
37380 
37407 
37434 
37461 



93358 

93348 

93337 

93327 

93316 

93306 

93295 

93285 

93274 

93264 

93253 

93243 

93232 

93222 

93211 

93201 

93190 

93180 

b'3169 

93159 

93148 

93137 

93127 

93116 

93106 

93095 

93084 

93074 

93063 

93052 

93042 

93031 

93020 

93010 

92999 

92988 

92978 

92967 

92956 

92945 

92935 

92926 

92913 

92902 

92892 

92881 

92870 



92849 
92838 
92827 
92816 
92805 
92794 
92784 
92773 
92762 
92751 
92740 
92729 
92718 



Tang. 



N. cos, 



iN.Sine 



60 

59 
B8 

57 
56 

55 
54 
58 
52 
51 
50 
49 

4b 
47 

46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
81 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 




68 Degrees. 



Log. Sines and Tangents. (22°) Natural Sines. 



43 



S ine. P. 10" Cosine. D. 10" Tang 



D. 10" Cotang. I N. sine.jN. cos. 



10 
11 
13 

13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
2S 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



.573575 
573888 
574200 
574512 
574824 
575136 
575447 
575758 
576069 
576379 
576689 

.576999 
577309 
577618 
577927 
578236 
578545 
578853 
579162 
579470 
579777 

.580085 
580392 
580699 
581005 
581312 
581618 
581924 
582229 
582535 
582840 

.583145 
583449 
583754 
584058 
584361 
584665 
584968 
585272 
585574 
585877 

.586179 
586482 
586783 
587085 
587386 
587688 
587989 
588289 
588590 
588890 

.589190 
589489 
589789 
590088 
590387 




52.1 
52.0 
52.0 
51.9 
51.9 
51.9 
51.8 
51.8 
51.7 
51.7 
51.6 
51.6 
51 .'6 
51.6 
51.5 
51.4 
51.4 
51.3 
51.3 



51.1 



51.1 
51.0 
51.0 
50.9 
50.9 
50.9 
50.8 
50.8 
50.7 
60.7 
50.6 
50.6 
50.6 
50.5 
50.5 
50.4 
50.4 
50.3 
50.3 
50.3 
50.2 
50.2 
50.1 
50.1 
50.1 
50.0 
50.0 
49.9 
49.9 
49.9 
49.8 
49.8 
49.7 
49.7 
49.7 
49.6 



9.967166 
967115 
967064 
967013 
966961 
966910 
966859 
966808 
966756 
966705 
966653 

9.966602 
966550 
966499 
966447 
966395 
966344 
966292 
966240 



966136 
966085 
966033 
965981 
965928 
965876 
965824 
965772 
965720 
965668 
965615 
9.965563 
965511 
965458 
965408 
965353 
965301 
965248 
965195 
965143 
965090 
965037 
964984 
964931 
964879 
964826 
964773 
964719 
964666 
964613 
964560 
964507 
964454 
964400 
964347 
964294 
964240 
964187 
964133 
964080 
964026 



8.5 
8.5 
8.6 
8.6 
8.5 
8.5 
8.5 
P. 5 
8.6 



8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 



8.6 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.8 
8.8 



8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 



9.606410 
606773 
607137 
607500 
607863 
608225 
608588 
608950 
609312 
609674 
610036 

9.610397 
610759 
611120 
611480 
611841 
612201 
612561 
612921 
613281 
613641 

9.614000 
614359 
614718 
615077 
615435 
615793 
616151 
616509 
616867 
617224 

9.617582 
617939 
618295 
618652 
619008 
619364 
619721 
620076 
620432 
620787 

9.621142 
621497 
621852 
622207 
622561 
622915 
623269 
623623 
623976 
624330 

9.624683 
625036 
625388 
625741 
626093 
626445 
626797 
627149 
627501 
627852 
Cotang. 



60 

60 

GO 

00 

60 

00 

60 

60.3 

60.3 

60.3 

60.2 

60.2 

60.2 

60.1 

60.1 

60.1 

60.0 

60.0 

60.0 

59.9 

59.9 

59.8 

59.8 

59.8 

59.7 

59.7 

59.7 

59.6 

59.6 

59.6 

59.5 

59.5 

59.5 

59.4 

59.4 

59.4 



59.3 
59.3 
59.3 
59.2 
59.2 
59.2 
59.1 
59.1 
59.0 
59.0 
59.0 
58.9 
58.9 
58.9 
58.8 
58.8 
58.8 
58.7 
58.7 
58.7 
58.6 
58.6 
58.6 
58.6 



10.393590 
393227 
392863 
392500 
392137 
391776 
391412i 
391050 
390688 
390326 
389964 

10.389603 
389241 
388880 
388520 
388159 
387799 
387439 
387079 
386719 
386359 

10-386000 
385641 
385282 
384923 
384565 
384207 
383849 
383491 
383133 
382776 

10-382418 
382061 
381705 
381348 
380992 
380636 
380279 
379924 
379568 
379213 

10-378858 
378503 
378148 
377793 
377439 
377085 
376731 
376377 
376024 
375670 

10-375317 
374964 
374612 
374259 
373907 
373555 
373203 
372861 
372499 
372148 
Tang. 



37461 
37488 
37515 
37542 
37569 
37595 
37622 
37649 
37676 
37703 
37730 
37757 
37784 
37811 
37838 
37865 
37892 
37919 
37946 
37973 
37999 
38026 
38053 
38080 
38107 
38134 
38161 
38188 
38215 
38241 



38295 
38322 
38349 
38376 
38403 
38430 
38456 
38483 
38510 
38537 
38564 
38591 
38617 
38644 
38671 



38725 
38752 
38778 



92718 
92707 
92697 
92686 
92675 
92664 
92653 
92642 
92631 
92620 
92609 
92598 
92587 
92576 
92565 
92554 
92543 
92532 
92521 
92510 
92499 
92488 
92477 
92466 
92455 
92444 
92432 
92421 
92410 
92399 
92388 
92377 
92366 
92355 
92343 
92332 
92321 
92310 
92299 
92287 
92276 
92265 
92254 
92243 
92231 
92220 
92209 
92198 
92186 
92175 



38805)92164 



38832 
38859 
38886 
38912 
38939 
38966 
38993 
39020 
39046 
39073 



92152 
92141 
92130 
92119 
92107 
92096 
92085 
92073 
92062 
92050 
N. co«. N.sine. 



(>! Degrees. 



19 



44 



Log. Sines and Tangents. (23°) Natural Sines. 



TABLE II. 




1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



598660 
598952 
599244 
599536 
599827 
600118 
600409 
600700 

1.600990 
601280 
601570 
601860 
602150 
602439 
602728 
603017 
603305 
603594 

.603882 
604170 
604457 
604745 
605032 
605319 
605606 
605892 
606179 
606465 

.606751 
607036 
607322 
607607 
607892 
608177 
608461 
608745 
609029 
609313 



D 10" 



Cosine. 



49.6 
49.5 
49.5 
49.5 
49.4 
49.4 
49.3 
49.3 



49 

49 

49 

49 

49 

49 

49 

49 

48 

48.9 

48.9 

48.8 

48.8 

48.7 

48.7 

48.7 

48.6 

48.6 

48.5 

48 

48 

48 

48 

48 

48 

48.3 

48.2 

48.2 

48.2 

48.1 

48.1 

48.1 

48.0 

48.0 

47.9 

47.9 

47.9 

47-8 

47.8 

47.8 

47.7 

47.7 

47.6 

47-6 

47-6 

47-5 

47-5 

47-4 

47.4 

47.4 

47.3 

47.3 



Cosine 

>. 964026 
963972 
963919 
963865 
963811 
963757 
963704 
963650 
963596 
963542 
963488 

(.963434 
963379 
963325 
963271 
963217 
963163 
963108 
963054 
962999 
962945 

• .962890 
962836 
962781 
962727 
962672 
962617 
962562 
962508 
962453 
962398 

.962343 
962288 
962233 
962178 
962123 
962067 
962012 
961957 
961902 
961846 

.961791 
961735 
961680 
961624 
961569 
961513 
961458 
961402 
961346 
961290 

.961235 
961179 
961123 
961067 
961011 
960955 
960899 
960843 
960786 
960730 



Sine. 



8.9 
8.9 
9.0 
9.0 
9.0 
9.0 
9.0 
9.0 



9 
9, 
9 
9 
9, 
9, 

9.0 

9.0 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

1 

2 

2 

2 

2 

2 

2 



9.2 
9.2 
9.2 
9.2 
9.2 
9.3 
9.3 
9.3 
9.3 



9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.4 
9.4 



Tang. 

9.627852 
628203 
628554 
628905 
629255 
629606 
629956 
630306 
630656 
631005 
631355 

9.631704 
632053 
632401 
632750 



633447 
633795 
634143 
634490 
634838 

9.635185 
635532 
635879 
636226 
636572 
636919 
637265 
637611 
637956 
638302 

9.638647 
638992 
639337 
639682 
640027 
640371 
640716 
641060 
641404 
641747 

9.642091 
642434 
642777 
643120 
643463 
643806 
644148 
644490 
644832 
645174 

9.645516 
645857 
646199 
646540 
646881 
647222 
647562 
647903 
648243 



l). 10" 



58.5 
58.5 
58.5 
58.4 
58.4 
58.3 



58. 



Cotang. 
Degrees. 



58.2 

58.2 

58.1 

58.1 

58.1 

58.0 

58.0 

58.0 

57.9 

57.9 

57.9 

57.8 

57.8 

57 

57.7 

57.7 

57.7 

57.7 

57.6 

57.6 

57.6 

57.5 

57.5 

57.5 

57.4 

57.4 

57.4 

57.3 

57.3 

57.3 

57.2 

57.2 

57.2 

57.2 

57.1 

57.1 

57.1 

57.0 

57.0 

57.0 

56.9 

56.9 

56.9 

56.9 

56.8 

56.8 

56.8 

56.7 

56.7 

56.7 



Cotang. 



sine. N. cos. 



10 



10 



10 



10 



10 



.372148 
371797 
371446 
371095 
870745 
370394 
370044 
S69694 
369344 
368995 
368645 
.368296 
367947 
367599 
367250 
366902 
366553 
366205 
365857 
365510 
365162 
.364815 
364468 
364121 
363774 
363428 
363081 
362735 
362389 
362044 
361698 
,361353 
361008 
360663 
360318 
359973 
359629 
359284 
358940 
358596 
358253 
357909 
357566 
357223 
356880 
356537 
356194 
355852 
355510 
355168 
354826 
354484 
354143 
353801 
353460 
353119 
352778 
352438 
352097 
351757 
351417 



Tang. 



39073 
39100 
39127 
39153 
39180 
39207 
39234 
39260 
39287 
39314 
39341 
39367 
i 39394 
39421 
39448 
39474 
39501 
39528 
39555 
39581 
39608 
39636 
39661 
39688 
39715 
39741 
39768 
39795 



39848 
39875 
39902 
39928 
39955 



40008 
40035 
40062 
40088 
40116 
40141 



40168 91578 



40195 
40221 
40248 
40275 
40301 
40328 
40356 
40381 
40408 
40434 
40461 
40488 
40514 
40541 
40567 
40594 
40621 
40647 
40674 



92050 
92039 
92028 
92016 
92005 
91994 
91982 
91971 
91959 
91948 
91936 
91925 
91914 
91902 
91891 
91879 
91868 
91856 
91845 
91833 
91822 
91810 
91799 
91787 
91775 
91764 
91752 
91741 
91729 
91718 
91706 
91694 
91683 
91671 
91660 
91648 
91636 
91625 
91613 
91601 
91590 



91566 
91555 
91543 
91531 
91519 
91508 
91496 
91484 
91472 
91461 
91449 
91437 
91425 
91414 
91402 
91390 
91378 
91366 
91365 



N. cos. N.sine. 



TABLE II. 



Log. Sines and Tangents. (24°) Natural 



45 




1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

25 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 



Sine. 

9.609313 
609597 
609880 
610164 
610447 
610729 
611012 
611294 
611576 
611858 
612140 

9.612421 
612702 
612983 
613264 
613545 
613825 
614105 
614385 
614665 
614944 

9.615223 
615502 
615781, 
616060 
616338 
616616 
616894 
617172 
617450 
617727 
9.618004 
618281 
618558 
618834 
619110 
619386 
619662 
619938 
620213 



D. 10 



47.3 
47.2 
47.2 
47.2 



9.620763 
621038 
621313 
621587 
621861 
622135 
622409 
622682 
622956 
623229 

9.623512 
623774 
624047 
624319 
624591 
624863 
625135 
625406 
625677 
625948 



47 

47 

47 

47 

47 

46 

46.9 

46.9 

46.8 

46.8 

46.7 

46.7 

46.7 

46.6 

46.6 

46.6 

46.5 

46.5 

46.5 

46.4 

46.4 

46.4 

46.3 

46.3 

46.2 

46.2 

46.2 

46.1 

46.1 

46.1 

46.0 



Cosine. 

.960730 
960674 
960618 
960561 
960505 
960448 
960392 
960335 
960279 
960222 
960165 
.960109 
960052 
959995 
959938 



Cosine. 



45.7 

45.7 

45.7 

45.6 

45.6 

45.6 

45.5 

45.5 

45.5 

45 

45 

45 

45 

45 

46 

45 

45 

45 



959825 
959768 
959711 
959654 
959596 
9.959539 
959482 
959425 
959368 
959310 
959253 
959195 
959138 
959081 
959023 
958965 



D. 10' 



958850 
958792 
958734 
958677 
958619 
958561 
958503 
958445 
.958387 
958329 
958271 
958213 
958154 
958096 
958038 
957979 
957921 
a57863 
9.957804 
957746 
957687 
957628 
957570 
957611 
957452 
957393 
957335 
957276 



9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.5 
9.5 
9.5 
9.5 
9.5 
9.5 
9.5 
9.5 
9.5 



9.5 
9.5 
9.5 
9.5 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 



Taiii* 



Sine. 



9.6 
9.6 
9.6 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 



.648583 

648923 

649263 

649602 

649942 

650281 

650620 

650959 

651297 

651636 

651974 

.652312 

652650 

652988 

653326 

653663 

654000 

654337 

654174 

655011 

655348 
9.655684 

656020 

656356 

656692 

657028 

657364 

657699 

658034 

658369 

658704 
9.659039 

659373 „ 

659708 £?'' 

660042 ™- 7 

660376 8r ' 

660710 °°*A 



D. 10 



56.6 
56.6 
56.6 
56 6 
56.5 
56.5 

.5 
56.4 
56.4 
56.4 
56.3 
56.3 
56.3 
56.3 
56.2 

.2 
56.2 
56.1 
56.1 
56.1 
56.1 
56.0 
56.0 
56.0 
55.9 
55.9 
55.9 
55.9 
55.8 
55.8 
55.8 
55.8 



Uotang. 



661043 
661377 
661710 
662043 
662376 
662709 
663042 
663375 
663707 
664039 
664371 
664703 
665035 
665366 
9.665697 
666029 
666360 
666691 
667021 
667352 
667682 
668013 
668343 
668672 



Cotang. 



65.6 
55.6 
55.6 
55.5 
55.5 
55.5 
55.4 
55.4 
55.4 
55.4 
65.3 
65.3 
55.3 
65.3 
55.2 
65.2 



65.2 
65.1 
56.1 
55.1 
55.1 
55.0 
55.0 
65.0 



10.351417 
351077 
350737 
360398 
350058 
349719 
349380 
349041 
348703 
348364 
348026 

10.347688 
347350 
347012 
346674 
346337 
346000 
345663 
345326 
344989 
344652 

10.344316 
343980 
343644 
343308 
342972 
342636 
342301 
341966 
341631 
341296 

10.340961 
340627 
340292 
339958 
339624 
339290 
338957 
338623 
338290 
337957 

10.337624 
337291 
336958 
336625 
336293 
335961 
335629 
335297 
334965 
334634 
10.334303 
333971 
333620 
333309 
332979 
332648 
332318 
331987 
331657 
331328 



Ta 



IN. I 



40674 
40700 
40727 
40753 
40780 



4080691295 



40833 



40886 
40913 
40939 
40966 



41019 
41045 
41072 
41098 
41125 
41161 



41178 91128 



41204 
41231 

41257 
41284 



N. cos. 



91355 
91343 
91331 
91319 
91307 



91283 
91272 
91260 
91248 
91236 
91224 



40992 91212 



91200 
91188 
91176 
91164 
91152 
91140 



4131091068 



41337 

41363 

41390 

41416 

41443 

41469 

41496 

41522 

41549 

41575 

41602 

41628 

41655 

41681 

41707 

41734 

41760 

41787 

41813 

41840 

41866 

41892 

41919 

41945 

41972 

41998 

42024 

42051 

42077 

42104 

42130 

42156 

42183 

42209 

42235 

42262 



91116 
91104 
91092 
91080 



91056 

91044 

91032 

91020 

91008 

90996 

90984 

90972 

90960 

90948 

90936 

90924 

90911 

90899 

90887 

90875 

90863 

90851 

90839 

90826 

90814 

90802 

90790 

90778 

90766 

90753 

90741 

90729 

90717 

90704 

90692 

90680 

90668 

90665 

90643 

90631 



N. cos. N.sine, 



65 Degrees. 



TABLE II. 



Log. Sines and Tangents. (26°) Natural Sines. 



47 



Sine. 



>. 641842 
642101 
642360 
642618 
642877 
643135 
643393 
643650 
643908 
644165 
644423 

1.644680 
644936 
645193 
645450 
645708 
645962 
646218 
648474 
646729 
646984 

1.647240 
647494 
647749 
648004 
648258 
648512 
648766 
649020 
649274 
649527 

1.649781 
650034 
650287 
650539 
650792 
651044 
651297 
651549 
651800 
652052 

'.652304 
652555 
652806 
653057 
653308 
653558 
653808 
654059 
654309 
654558 

.654808 
655058 
655307 
655556 
655805 
656054 
656302 
656551 
656799 
657047 

Cosine. 



D. 10" 



43.1 

43.1 

43.1 

43.0 

43.0 

43.0 

43.0 

42.9 

42.9 

42.9 

42.8 

42.8 

42.8 

42 

42 

42 

42 

42 

42 

42 

42.5 

42.5 

42.4 

42.4 

42.4 

42 

42 

42 

42 

42 

42 

42 

42.2 

42.1 

42.1 

42.1 

42.0 

42.0 

42.0 

41.9 

41.9 

41.9 

41.8 

41.8 

41.8 

41.8 

41.7 

41.7 

41.7 

41.6 

41.6 

41.6 

41.6 

41.5 

41.5 

41.5 

41.4 

41.4 

41.4 

41.3 



Cosine. 



.953660 
953599 
953537 
953475 
953413 
953352 
953290 
953228 
953166 
953104 
953042 

.952980 
952918 
952855 
952793 
952731 
952669 
952606 
952544 
952481 
952419 

.952356 
952294 
952231 
952168 
952106 
952043 
951980 
951917 
951854 
951791 

.951728 
951665 
951602 
951539 
951476 
951412 
951349 
951286 
951222 
951159 
951096 
951032 



950905 
950841 
950778 
950714 
950650 
950586 
950522 
. 950458 
950394 
950330 
950366 
950202 
950138 
950074 
950010 
949945 
949881 



D. 10" 



10.4 



10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10.5 

10.5 

10.6 

10.5 

10.5 

10.5 

10.5 

10.5 

10.5 

10.5 

10.5 

10.5 

10.6 

10 

10 

10 

10 

10 

10 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.7 

10 

10 

10 

10 

10 

10 

10 

10 



10.7 



Tang. 

.688182 
688502 
688823 
689143 
689463 
689783 
690103 
690423 
690742 
691062 
691381 

.691700 
692019 
692338 
692656 
692975 
693293 
693612 
693930 
694248 
694566 

.694883 
695201 
695518 
695836 
696153 
696470 
696787 
697103 
697420 
697736 

.698053 
698369 
698685 
699001 
699316 
699632 
699947 
700263 
700578 
700893 

, 701208 
701523 
701837 
702152 
702468 
702780 
703095 
703409 
703723 
704036 
704350 
704663 
704977 
705290 
705603 
705916 
706228 
706541 
706854 
707166 



D. 10"! Cotanj 



53.4 
53.4 
53.4 
53.3 



53 

53 

53 

53 

53 

53 

53 

53 

53.1 

53.1 

53.1 

53.1 

53.0 

53.0 

63.0 

53.0 

52.9 

52.9 

52.9 

62.9 

52.9 

52.8 

52.8 

52.8 

52.8 

52.7 

52.7 

52.7 

52.7 

52.6 

62.6 

52.6 

52.6 

52.6 

52.5 

52.5 

52.5 

52.4 

52.4 

52.4 

52.4 

52.4 



52 

52 

52.2 

52.2 

52.2 

52.1 

52.1 

52.1 

52.1 

52.1 



Cctan: 



10.311818 
311498 
311177 
310857 
310537 
310217 
309897 
309577 
309258 
308938 
308619 

10.308300 
307981 
307662 
307344 
307025 
306707 



306070 
305752 
305434 

10-305117 
304799 
304482 
304164 
303847 
303530 
303213 
302897 
302580 
302264 

10-301947 
301631 
301315 
300999 
300684 
300368 
300053 
299737 
299422 
299107 

10-298792 
298477 
298163 
297848 
297534 
297220 
296905 
296691 
296277 
295964 

10-295650 
295337 
295023 
294710 
294397 
294084 
293772 
293459 
293146 
292834 



Tang. 



;N. sine. N. cos, 

89879 
89867 
89854 
89841 
89828 
89816 



4383 1 
43863 
43889 
| 43916 
! 43942 
43968 
43994 
44020 
44046 
44072 
44098 
44124 
44151 
44177 
44203 
44229 
44255 
44281 
44307 
44333 
44359 
44385 
44411 
44437 
44464 
44490 
44516 
44542 
44568 
44594 
44620 
44646 
44672 
44698 
44724 
44750 
44776 
44802 



44854 
44880 
44906 
44932 
44958 
44984 
45010 
45036 
45062 
45088 
45114 
45140 
45166 
45192 
45218 
45243 
45269 
45295 



89790 
89777 
89764 
89752 
89739 
89726 
89713 
89700 
89687 
89674 
89662 
89649 
89636 
89623 
89610 
89597 
89584 
89571 
89558 
89545 
89532 
89519 
89506 
89493 
89480 
89467 
89454 
89441 
89428 
89415 
89402 
89389 
89376 
89363 
89350 
89337 
89324 
89311 
89298 
89286 
89272 
89259 
89245 
89232 
89219 
89208 
89193 
S9180 
89167 
89153 



4532189140 
4534789127 
45373189114 
46399 89101 

N. cos. |N. sine. 



63 Degrees. 



48 



Log. Sines and Tangents. (27°) Natural Sines. 



TABLE II. 




1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 

.657047 
657295 
657542 
657790 
658037 
658284 
658531 
658778 
659025 
659271 
659517 

.659763 
660009 
660255 
660501 
660746 
660991 
661236 
661481 
661726 
661970 

.662214 
662459 
662703 
662946 
663190 
663433 
663677 
663920 
664163 
664406 

.664648 
664891 
665133 
665375 
665617 
665859 
666100 
666342 
666583 
666824 

.667065 
667305 
667546 
667786 
668027 
668267 
668506 
668746 




D. 10'' 



41.3 
41.3 

41.2 
41.2 
41.2 
41.2 
41.1 
41.1 
41.1 
41.0 
41.0 
41.0 
40.9 
40.9 
40.9 
40.9 
40.8 
40.8 
40.8 
40.7 
40.7 
40.7 
40.7 
40.6 
40.6 
40.6 
40.5 
40.5 



40 

40 

40 

40 

40 

40 

40 

40 

40 

40 

40 

40 

40 

40 

40 

40 

40.0 

40 

40.0 

39.9 

39.9 

39.9 

39.9 

39.8 

39.8 

39.8 

39.7 

39.7 

39.7 

39.7 

39.6 

39.6 



Cosine. 

.949881 
949816 
949752 
949688 
949623 
949558 
949494 
949429 
949364 
949300 
949235 

.949170 
949105 
949040 
948975 
948910 
948845 
948780 
948715 
948650 
948584 

.948519 
948454 
948388 
948323 
948257 
948192 
948126 
948060 
947995 
947929 

.947863 
947797 
947731 
947665 
947600 
947533 
947467 
947401 
947335 
947269 

.947203 
947136 
947070 
947004 
946937 
946871 
946804 
946738 
946671 
946604 

.946538 
946471 
946404 
946337 
946270 
946203 
946136 
946089 
946002 
945935 
Sine! 



D. lu 



10.7 

10.7 

10.7 

10.8 

10.8 

10.8 

10.8 

10.8 

10.8 

10.8 

10.8 

10.8 

10.8 

10.8 

10.8 

10.8 

10.8 

10.9 

10.9 

10.9 

10.9 

10.9 

10.9 

10.9 

10.9 

10.9 

10.9 

10.9 

10.9 

11.0 

11.0 

11.0 

11.0 

11.0 

11.0 

11.0 

11.0 

11.0 

11.0 

11.0 

11.0 

11.0 

11. 

11. 

11. 

11. 

11. 

11. 

11. 

11. 

11. 

11. 

11. 

11. 

11. 

11.2 

11.2 

11.2 

11.2 

11.2 



Taus. il». lu 



9. 



707166 

707478 

707790 

708102 

708414 

708726 

709037 

709349 

709660 

709971 

710282 

710593 

710904 

711215 

711525 

711836 

712146 

712456 

712766 

713076 

713386 

713696 

714005 

714314 

714624 

714933 

715242 

715551 

715860 

716168 

716477 

716785 

717093 

717401 

717709 

718017 

718325 

718633 

718940 

7192481 

719555 

719862 

720169 

720476 

720783 

721089 

721396 

721702 

722009 

722315 

722621 

722927 

723232 

723538 

723844 

724149 

724454 

724759 

725065 

725369 

725674 



52.0 
52.0 
52.0 
52.0 
51.9 
51.9 
51.9 
51.5 
51.9 
51.8 
51.8 
51.8 
51.8 
61.8 
51.7 
61.7 
51.7 
51.7 
61.6 
51.6 
51.6 
51.6 
51.6 
51.5 
51.5 
51.5 
51.5 
51.4 
51.4 
51.4 
51.4 
51.4 
51.3 
51.3 
51.3 
51.3 
51.3 



51 

51 

51 

51 

51 

51 

51 

51 

51.1 

51.1 

51.0 

51.0 

51.0 

51.0 

51.0 

50.9 

50.9 

50.9 

50.9 

50.9 

50.8 

60.8 

50.8 



Cotang. 



Cotang. | N. sine. N. 



10.292834 
292522 
292210 
291898 
291686 
291274 
290963 
290651 
290340 
290029 
289718 

10.289407 
289096 
288785 
288475 
288164 
287854 
287544 
287234 
286924 
286614 

10.286304 
285995 
285686 
286376 
286067 
284768 
284449 
284140 
283832 
283523 

10.283215 
282907 
282599 
282291 
281983 
281676 
281367 
281060 
280762 
280445 

10.280138 
279831 
279524 
279217 
278911 
278604 
278298 
277991 
277685 
277379 

16 277073 
276768 
276462 
276166 
275851 
275546 
275241 
274935 
274631 
274326 



Tang. 



45399 
45425 
45451 
45477 
45503 
45529 
45554 
45580 
45606 
45632 
45658 
45684 
45710 
45736 
45762 
45787 
45813 
45839 
45865 
45891 
45917 
45942 
46968 
45994 
46020 
46046 
46072 
46097 
46123 
46149 
46175 
46201 
46226 
46252 
46278 
46304 
46330 
46355 
46381 
46407 
46433 
46458 
46484 
46510 
46636 
46561 
46587 
46613 
46639 
46664 
46690 
46716 
46742 
46767 
46793 
46819 
46844 
46870 
46896 
46921 
46947 



N. cos. iN.srn 



89101 
89087 
89074 
89061 
89048 
89035 
89021 
89008 
88995 
88981 
88968 
88955 
88942 
88928 
88915 
88902 
88888 
88875 
88862 



88822 
308 
795 
88782 
88768 
88755 
88741 
88728 
88715 
88701 
388 
374 
88661 
88647 
88G34 
88620 
88607 
38593 
88580 
88566 
88553 
88539 
88526 
88512 
88499 
88485 
88472 
88458 
88445 
88431 
88417 
88404 
88390 
88377 
88363 



88322 
88308 
88295 



62 Degrees. 



Log. Sines and Tangents. (28°) Natural Sines. 



49 



ine. D. 10" Cosine. D. 10" Tang. ID. 10" Cotang. ,N. sine. N. cos 



10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

2G 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

33 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 |9 

52 

53 

54 

55 

56 

57 

58 

59 

CO 



.671609 
671847 
67208*4 
672321 
672558 
672795 
673032 
673268 
673505 
673741 
673977 

. 674213 
674448 
674684 
674919 
675155 
675390 
675S24 
675859 
676094 
676328 

.676562 
676795 
677030 
677264 
677498 
677731 
677964 
678197 
678430 
678663 

.678895 
679128 
679360 
679592 
679824 
680056 
680288 
680519 
680750 
680982 

.681213 
681443 
681674 
681905 
682135 
682365 
682595 
682825 
683055 
683284 

.683514 
683743 
683972 
684201 
684430 
684658 
684887 
685115 
685343 
685571 



Cosine. 



39.5 
39.5 
39.5 
39.5 
39.4 
39.4 
39.4 



39 
39 
39 
39 
39 
39 
39 

39.2 
39.1 
39.1 
39.1 
39.1 
39.0 
39.0 
39.0 
39.0 
38.9 
38.9 
38.9 
38.8 
38.8 
38.8 
38.8 
38.7 
38.7 
38.7 
7 
6 
6 
6 
5 
5 
5 
5 
4 
38.4 
38.4 
38.4 
38.3 
38.3 
38.3 
38.3 
38.2 
38.2 
38.2 
38.2 
38.1 
38.1 
38.1 
38.0 
38.0 
38.0 



9.945935 
945868 
945800 
945733 
945666 
945598 
945531 
945464 
945396 
945328 
945261 

9.945193 
945125 
945058 
944990 
944922 
944854 
944786 
944718 
944650 
944582 

9.944514 
944446 
944377 
944309 
944241 
944172 
944104 
944036 
943967 
943899 
943830 
943761 
943693 
943624 
943555 
943486 
943417 
943348 
943279 
943210 
943141 
943072 
943003 
942934 
942864 
942795 
942726 
942656 
942587 
942517 

9.942448 
942378 
942308 
942239 
942169 
942099 
942029 
941959 
941889 
941819 



Sine. 



11.2 
11.2 
11.2 
11.2 
11.2 
11.2 
11.2 
11.3 
11.3 
11.3 
11.3 
11.3 
11.3 
11.3 
11.3 
11.3 
11.3 
11.3 
11.3 
11.3 
11.4 
11.4 
11.4 



11 

11 

11 

11 

11 

11 

11 

11 

11 

11.4 

11.5 

11.5 

11.5 

11.5 

11.5 

11 

11 



11 

11 

11 

11 

11 

11 

11.6 

11.6 

11.6 

11.6 

11.6 

11.6 

11.6 

11.6 

11.6 

11.6 

11.6 

11.6 

11.6 

11.7 



1.725674 
725979 
726284 
726588 
726892 
727197 
727501 
727805 
728109 
728412 
728716 

1.729020 
729323 
729626 
729929 
730233 
730535 
730838 
731141 
731444 
731746 

•.732048 
732351 
732653 
732955 
733257 
733558 
733860 
734162 
734463 
734764 

.735066 
735367 
735668 
735969 
736269 
736570 
736871 
737171 
737471 
737771 

.738071 
738371 
738671 
738971 
739271 
739570 
739870 
740169 
740468 
740767 

.741066 
741365 
741664 
741962 
742261 
742559 
742858 
743156 
743454 
743752 



Cotang. 



50.8 
50.8 
50.7 
50.7 
50.7 



50 

50 

50 

50 

50 

50 

50 

50 

50 

50 

50 

50 

50 

50 

50-4 

50.4 

50-4 

50-3 

50-3 

50.3 

50-3 

50-3 

50-2 

50-2 

50-2 

50-2 

50.2 

50.2 

50.1 

50.1 

50.1 

50.1 

50.1 

50.0 

50.0 

50.0 

50.0 

50.0 

49.9 

49.9 

49.9 

49.9 

49.9 

49.9 

49.8 

49.8 

49.8 

49.8 

49.8 



10.274326 
274021 
273716 
273412 
273108 
272803 
272499 
272195 
271891 
271588 
271284 

10.270980 
270677 
270374 
270071 
269767 
269465 
269162 



268556 
268254 

10.267952 
267649 
267347 
267045 
266743 
266442 
266140 
265838 
265537 
265236 

10.264934 
264633 
264332 
264031 
263731 
263430 
263129 
262829 
262529 
262229 

10.261929' 
261629 | 
261329 I 
261029 ' 
260729 
260430 
260130 
259831 
259532 
259233 

10.258934 
258635 
258336 
258038 
257739 
257441 
257142 
256844 



256546 
256248 



46947 
46973 
46999 
47024 
47050 
47076 
47101 
47127 
47153 
47178 
47204 
47229 
47255 
47281 
47306 
47332 
47358 
47383 
47409 
47434 
47460 
47486 
47511 
47537 
47562 
47588 
47614 
47639 
47665 
47690 
47716 
j 47741 
47767 
! 47793 
j 47818 
47844 
47869 
47895 
47920 
47946 
47971 
47997 
48022 
48048 
48073 
48099 
48124 
48150 
48175 
| 48201 
48226 
48252 
48277 
[ 48303 
j 48328 
48354 
! 48379 
i 48405 
! 48430 
! 48456 
: 48481 



88295 
88281 
88267 
88254 
88240 
88226 
88213 
88199 
88185 
88172 
88158 
88144 
88130 
88117 
88103 
88089 
88075 
88062 
88048 
88034 
88020 
88006 
87993 
87979 
87965 
87951 
87937 
87923 
87909 
87896 
87882 
87868 
87854 
87840 
87826 
87812 
87798 
87784 
87770 
87756 
87743 
87729 
87715 
87701 
87687 
87673 
87659 
87645 
87631 
87617 
87603 
87589 
87575 
87561 
87546 
87532 
518 
504 
490 
87476 
87462 



N. cos.|N.sine. 



50 



Log. Sines and Tangents. 



J°) Natural Sines. 



TABLE II. 



S ine. P. 10" Cosi ne. | D. 10" 




1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



690098 

.690323 
690548 
690772 
690996 
691220 
691444 
691668 
691892 
692115 
692339 

.692562 
692785 
693008 
693231 
693453 
693676 
693898 
694120 
694342 
694564 

.694786 
695007 
695229 
695450 
695671 
695892 
696113 
696334 
696554 
696775 

. 696995 
697215 
697435 
697654 
697874 
698094 
698313 
698532 
698751 
698970 
Cosine. 



38.0 
37.9 
37.9 
37.9 
37.9 
37.8 
37.8 
37.8 
37.8 
37.7 
37.7 
37.7 
37.7 
37.6 
37.6 
37.6 
37.6 
37.5 
37.5 
37.5 
37.5 
37.4 
37.4 
37.4 
37.4 
37.3 
37.3 
37.3 
37.3 
37.2 
37.2 
37.2 
37.1 
37.1 
37.1 
37.1 
37.0 
37.0 
37.0 
37.0 
36.9 
36.9 
36.9 
36.9 
36.8 
36.8 
36.8 
36.8 
36.7 
36.7 
36.7 
36.7 
36.6 
36.6 
36.6 
36.6 
36.5 
36.5 
36.5 
36.5 



.941819 
941749 
941679 
941609 
941539 
941469 
941398 
941328 
941258 
941187 
941117 
.941046 
940975 
940905 
940834 
940763 
940693 
940622 
940551 
940480 
940409 
.940338 
940267 
940196 
940125 
940054 
939982 
939911 
939840 
939768 
939697 
.939625 
939554 
939482 
939410 
939339 
939267 
939195 
939123 
939052 
938980 
.938908 
938836 
938763 
938691 
938619 
938547 
938475 
938402 
938330 
938268 
.938185 
938113 
938040 
937967 
937895 
937822 
937749 
937676 
937604 
937531 



Sine. 



11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 



Tans. 



D. 10" 



1.743752 
744050 
744348 
744645 
744943 
745240 
745538 
745835 
746132 
746429 
746726 

.747023 
747319 
747616 
747913 
748209 
748505 
748801 
749097 
749393 
749689 

.749985 
750281 
750576 
750872 
751167 
751462 
751757 
752052 
752347 
752642 

.752937 
753231 
753526 
753820 
754115 
764409 
754703 
754997 

"755291 
755685 

.755878 
756172 
756465 
756759 
757052 
757345 
757638 
757931 
758224 
758517 

.758810 
759102 
759395 
759687 
759979 
760272 
760564 
760856 
761148 
761439 



Cotang. 



Cotang. 

10.256248 
255950 
255652 
255355 
255057 
254760 
254462 
254165 
253868 
253571 
253274 

10.252977 
252681 
252384 
252087 
251791 
251495 
251199 
250903 
250607 
250311 

10.250015 
249719 
249424 
249128 
248833 
248538 
248243 
247948 
247653 
247358 

10.247063 
246769 
246474 
246180 
245885 
245591 
245297 
245003 
244709 
244415 

10.244122 
243828 
243536 
243241 
242948 
242655 
242362 
242069 
241776 
241483 

10.241190 
240898 
240605 
240313 
240021 
239728 
239436 
239144 
238852 , 
238561 



N. sine. N. cos 



48481 
48506 
48532 
48557 
48583 



48634 



48659 87363 



48684 
48710 
48735 
48761 
48786 
48811 
48837 
48862 



48913 



48964 



87462 
87448 
87434 
87420 
87406 
87391 
87377 



49014 
49040 
49065 
49090 
49116 
49141 
49166 
49192 
49217 
49242 
49268 
49293 
49318 
49344 
49369 
49394 
49419 
49445 
49470 
49495 
49521 
49546 
49571 
49S96 
49622 
49647 
49672 
49697 
49723 
49748 
49773 
49798 
49824 
49849 
49874 



49924 
49950 
49975 
50000 



87349 
87335 
87321 
87306 
87292 
87278 
87264 
87250 
87235 
87221 
87207 
87193 
87178 
87164 
87150 
87136 
87121 
87107 
87093 
87079 

7064 
87050 

7036 
87021 
87007 
86993 
86978 
86964 
86949 
86935 
86921 
86908 
86892 
86878 
86863 
86849 
86834 
86820 
86805 
86791 
86777 
86762 
86748 
86733 
86719 
86704 
86690 
86675 
86661 
86646 
86632 
86617 
86603 



Tang. 



N. cos. IV. mn\ 



60 Degrees. 



Log. Sines and Tangents. (30°) Natural Sines. 



51 



10 
11 
12 

13 
14 
15 
16 
1? 
18 
19 
20 
21 
22 
23 
24 
25 
26 
2? 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
33 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



9.698970 

699189 

699407 

699626 

699844 

700062 

700280 

700498 

700716 

700933 

701151 

9.701368 

701585 

701802 

702019 

702236 

702452 

702669 

702885 

703101 

703317 

9.703533 

703749 

703964 

704179 

704395 

704610 

704825 

705040 

705254 

705469 

9.705683 

705898 

706112 

706326 

706539 

706753 

706967 

707180 

707393 

707606 

9.707819 

708032 

708245 

708458 

708670 

708882 

709094 

709306 

709518 

709730 

9.709941 

710153 

710J64 

710575 

710786 

710967 

711208 

711419 

711629 

711839 

Cosine. | 



36.4 


36.4 


38.4 


36.4 


36.3 


136.3 


36.3 


36.3 


36.3 


36.2 


36.2 


36.2 


36.2 


36.1 


36.1 


36.1 


36.1 


36.0 


36.0 


36.0 


36.0 


35.9 


35.9 


35.9 


35.9 


35.9 


35.8 


35.8 


35.8 


35.8 


35.7 


35.7 


35.7 


35.7 


35.6 


35.6 


35.6 


35.6 


35.5 


35.5 


35.5 


35.5 


35.4 


35.4 


35.4 


35.4 


35.3 


35.3 


35.3 


35.3 


35.3 


35.2 


35.2 


35.2 


35.2 


35.1 


35.1 


35.1 


35.1 


35.0 



Cosine. 

.937531 
937458 
937385 
937312 
937238 
937165 
937092 
937019 
936946 
936872 
936799 
.936725 
936652 
936578 
936505 
936431 
936357 
936284 
936210 
936136 
936062 
.935988 
935914 
935840 
935766 
935692 
935618 
935543 
935469 
935395 
935320 
.935246 
935171 
935097 
935022 
934948 
934873 
934798 
934723 
934649 
934574 
•934499 
934424 
934349 
934274 
934199 
934123 
934048 
933973 
933898 
933822 
933747 
933671 
933596 
933520 
933445 
933369 
933293 
933217 
933141 
933066 



D. 10' 

12.1 
12.2 
12.2 
12.2 
12.2 
12.2 
12.2 
12.2 



Sine 



12 

12 

12 

12 

12 

12 

12 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 



r ran>. 

9.761439 
761731 
762023 
762314 
762606 
762897 
763188 
763479 
763770 
764061 
764352 

9.764643 
764933 
765224 
765514 
765805 
766095 
766385 
766675 
766965 
767255 
767545 
767834 
768124 
768413 
768703 
768992 
769281 
769570 
769860 
770148 

9.770437 
770726 
771015 
771303 
771592 
771880 
772168 
772457 
772745 
773033 
773321 
773608 
773896 
774184 
774471 
774759 
775046 
775333 
775621 
775908 

9.776195 
776482 
776769 
777055 
777342 
777628 
777915 
778201 
778487 
778774 



Cotanp 



[ N. sine. N. cos 



10.238561 
238269 
237977 
237686 
237394 
237103 
236812 
236521 
236230 
235939 
235648 

10.235357 
235067 
234776 
234486 
234195 
233905 
233615 
233325 
233035 i 
232745 

10.232455 
232166 
231876 
231587 
231297 
231008 
230719 
230430 
230140 
229852 

10-229563 
229274 
228985 
228697 
228408 
228120 
227832 
227543 
227255 
226967 

10-226679 
226392 
226104 
225816 
225529 
225241 
224954 
224667 
224379 
224092 

10-223805 
223518 
223231 
222945 
222658 
222372 
222085 
221799 
221512 
221226 
"Tang. 



50000 

50025 

50050 

50076 

50101 

50126 

50151 

50176 

50201 

50227 

50252 

50277 

50302 

50327 

50352 

50377 

50403 

50428 

50453 

50478 

50503 

50528 

50553 

50578 

50603 

50628 

50654 

50679 

50704 

50729 

50754 

50779 

50804 

50829 

50854 

5087 

50904 

50929 

50954 

50979 

51004 

51029 

51054 

51079 

51104 

61129 

51154 

51179 

51204 

51229 

51254 

51279 

51304 

51329 

51354 

51379 

51404 

51429 

51454 

51479 

51504 



86603 
86588 
86573 
86559 
36544 
36530 
86515 
86501 
86486 
86471 
86457 
86442 
86427 
86413 



86384 
86369 
86354 
86340 
86325 
86310 
86295 
86281 
86266 
86251 
86237 
86222 
86207 
86192 
86178 
86163 
86148 
86133 
86119 
86104 
86089 
86074 
86059 
86045 
86030 
86015 
86000 
85985 
85970 
85956 
85941 
85926 
85911 
85836 
85881 
85866 
85851 
85836 
85821 
85808 
85792 
85777 
85762 
85747 
85732 
85717 
N. cos. N.sine. 



59 Degrees. 



52 



Log. Sines and Tangents. (31°) Natural Sines. 



TABLE II. 




1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
67 
58 
59 
60 



1.711839 og 

712050 sr 

712260 2' 

712469 JJ- 

712679 2' 

712889 J*" 

713098 XT 

713308 2' 

713517 JJ" 

713726 If 

713935 2" 

(.714144 2' 

714352 2" 

714561 2' 

714769 2" 

714978 2' 

715186 2" 

715394 JJ' 

715602 2' 

715809 2" 

716017 2' 

..716224 2' 

716432 oT' 

716639 2' 

716846 2' 

717053 2" 

717259 XT' 

717466 2" 

717673 2" 

717879 2' 

718085 2' 

1.718291 2" 

718497 2" 

718703 2' 

718909 2' 

719114 2' 

719320 2' 

719525 2 

719730 2' 

719935 2 

720140 I 2' 

(.720345 2' 

720549 2' 

720754 2' 
720958; 2' 
721162! 2 
721366! 2 

721570 '2' 
721774 | 2 

721978; 2' 
722181(2 '. 
1.722385 2' 
722588 1 2". 
722791 j 2' 
722994 2'' 
723197; 2': 
723400 2 "• 
723603 2" 
723805 2* 
724007 2' 
724210 dd - 



D. 10" Conine. 

9.933036 
982990 
932914 
932838 
932762 
932685 
932609 
932533 
932457 
932380 
932304 

9.932228 
932151 
932075 
931998 
931921 
931845 
931768 
931691 
931614 
931537 

9.931460 
931383 
931306 
931229 
931152 
931075 
930998 
930921 
930843 
930766 

9.930688 
930611 
930533 
930456 
930378 
930300 
930223 
930145 
930067 



.929911 
929833 
929755 
929677 
929599 
929521 
929442 
929364 
929286 
929207 
.929129 
929050 
928972 
928893 
928815 
928736 
928657 
928578 
928499 
928420 



Sine. 



D. 10" 



12 

12 

12 

12 

12 

12 

12 

12 

12.7 

12.7 

12.7 

12.7 

12.7 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 



Tang. D/10" 



.778774 
779060 
779346 
779632 
779918 
780203 
780489 
780775 
781060 
781346 
781631 

.781916 
782201 
782486 
782771 
783056 
783341 
783626 
783910 
784195 
784479 

.784764 
785048 
785332 
785616 
785900 
786184 
786468 
786752 
787036 
787319 

.787603 
787886 
788170 
788453 
788736 
789019 
789302 
789585 
789868 
790151 

.790433 
790716 
790999 
791281 
791563 
791846 
792128 
792410 
792692 
792974 

.793256 
793538 
793819 
794101 
794383 
794664 
794945 
795227 
795508 
795789 



Cotang. 



47.7 

47.7 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.5 

47.5 

47.5 

47.5 

47.5 

47.5 

47.5 

47.5 

47.4 

47.4 

47.4 

47.4 

47.4 

47.4 

47.3 

47.3 

47.3 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47.1 

47.1 

47.1 

47.1 

47.1 

47.1 

47.1 

47.0 

47.0 

47.0 

47.0 

47.0 

47.0 

47.0 

46.9 

46.9 

46.9 

46.9 

46.9 

46.9 

46.9 

46.8 



Cotang. 



10.221226 
220940 
220654 
220368 
220082 
219797 
219511 
219225 
218940 
218654 
218369 

10.218084 
217799 
217514 
217229 
216944 
216659 
216374 
216090 
215805 
215521 

10.215236 
214952 
214668 
214384 
214100 
213816 
213532 
213248 
212964 
212681 

10.212397 
212114 
211830 
211547 
211264 
210981 
210698 
210416 
210132 
209849 

10.209567 
209284 
209001 
208719 
208437 
208154 
207872 
207590 
207308 
207026 

10.206744 
206462 
206181 
205899 
205617 
205336 
205055 
204773 
204492 
204211 




51952185446 
51977185431 
52002185416 
52026185401 
52051 185385 
52076|85370 



62101 
52126 
52151 
52176 



85355 
85340 
85325 
85310 



52200185294 
52225J85279 
5225085264 
52275185249 



52299 
52324 
52349 
52374 
52399 
52423 
52448 
52473 
52498 
52522 
52547 
52572 
52597 
52621 
52646 
52671 
62696 
62720 
52745 
152770 
1 52794 
152819 
i 52844 
52869 
52893 
52918 
62943 
52967 
52992 



Tang. 



85234 
85218 
85203 
85188 
85173 
85157 
85142 
85127 
85H2 
85096 
85081 
85066 
85051 
85036 
85020 
85005 
84989 
84974 
84959 
84943 
84928 
84913 
84897 
84882 
84866 
84851 
84836 
84820 
84805 



N. cos.'N.siue, 



58 Degrees. 



TABLE II. 



Log. Sines and Tangents. (32°) Natural Sines. 



53 



_ Sine. 

1.724210 
724412 
724614 
724816 
725017 
725219 
725420 
725622 
725823 
726024 
726225 

'.726426 
726626 
726827 
727027 
727228 
727428 
727628 
727828 
728027 
728227 

'.728427 
728626 
728825 
729024 
729223 
729422 
729621 
729820 
730018 
730216 

.730415 
730613 
730811 
731009 
731206 
731404 
731602 
731799 
731996 
732193 

.732390 
732587 
732784 
732980 
733177 
733373 
733569 
733765 
733961 
734157 

(.734353 
734549 
734744 
734939 
735135 
735330 
735525 
735719 
735914 
736109 
Cosine. 



D. 10' 



33.7 
33.7 
33.6 
33.6 
33.6 



33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33.3 

33.3 

33.3 

33.3 

33.3 

33.2 

33.2 

33.2 

33.2 

33.1 

33.1 

33.1 

33.1 

33.0 

33.0 

33.0 

33.0 

33.0 

32.9 

32.9 

32.9 

32.9 

32.9 

32.8 

32.8 

32.8 

32.8 

32.8 

32.7 

32.7 

32.7 

32.7 

32.7 

32.6 

32.6 

32.6 

32.6 

32.5 

32.5 

32.5 

32.5 

32.5 

32.4 

32.4 



Cosine. |D. 10' 



928420 
928342 
928263 
928183 
928104 
928025 
927946 
927867 
927787 
927708 
927629 

9.927549 
927470 
927390 
927310 
927231 
927151 
927071 
926991 
92691 1 
926831 

9.926751 
926671 
926591 
926511 
926431 
926351 
926270 
926190 
926110 
926029 
.925949 
925868 
925788 
925707 
925626 
925545 
925465 
925384 
925303 
925222 
.925141 
925060 
924979 
924897 
924816 
924735 
924654 
924572 
924491 
924409 

9.924328 
924246 
924164 
924083 
924001 
923919 
923837 
923755 
923673 
923591 



Sine. 



13.2 

13.2 

13.2 

13 

13 

13 

13 

13 

13 

13 

13.2 

13.2 

13.3 

13.3 

13.3 

13.3 

13.3 

13.3 

13.3 

13.3 

13.3 

13.3 



13.3 
13.3 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.5 
13.5 
13.5 
13.5 
13.5 
13.5 
13.5 
13.5 
13.5 
13.5 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.7 
13.7 



Tang. 



9.795789 
796070 
796351 
796632 
796913 
797194 
797475 
797755 
798036 
798316 
798596 

9.798877 
799157 
799437 
799717 
799997 
800277 
800557 
800836 
801116 
801396 

9.801675 
801955 
802234 
802513 
802792 
803072 
803351 
803630 
803908 
804187 

9.804466 
804745 
805023 
805302 
805580 
805859 
806137 
806415 
806693 
806971 

9.807249 
807527 
807805 
808083 
808361 
808638 
808916 
809193 
809471 
809748 

9.810025 
810302 
810580 
810857 
811134 
811410 
811687 
811964 
812241 
812517 
Cotang. 
57 Degrees. 



D. 10"j Cotang. I N. Sine.jJN 



46.8 

46.8 

46.8 

46.8 

46.8 

46.8 

46.8 

46.8 

46, 

46 

46 

46, 

46 

46 

46 

46, 

46 

46 



10. 



10. 



10. 



10. 



! 53041 

53066 
| 63091 
153115 
|53140!84712 
153164 

53189 

53214 

53238 

53263 

53288 

53312 

53337 

53361 

53386 

53411 

53435 

53460 

53484 

53509 

53534 

53558 

53583 

53607 

53632 



10. 



Tang. 



N. cos. N.siuo. 



54 



Log. Sines and Tangents. (33°) Natural Sines. 



TABLE n. 



Sine. 



10 
11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 



>. 736109 
736303 
736498 
736692 
736886 
737080 
737274 
737467 
737661 
737855 
738048 

(.738241 
738434 
738627 
738820 
739013 
739206 
739398 
739590 
739783 
739975 

'.740167 
740359 
740550 
740742 
740934 
741125 
741316 
741508 
741699 
741889 

'.742080 
742271 
742462 
742652 
742842 
743033 
743223 
743413 
743602 
743792 

.743982 
744171 
744361 
744550 
744739 
744928 
745117 
745306 
745494 
745683 

.745871 
746059 
746248 
746436 
746624 
746812 
746999 
747187 
747374 
747562 

Cosine. 



D. 10" 



32.4 
32.4 
32.4 
32.3 
32.3 
32.3 
32.3 
32.3 
32.2 
32.2 
32.2 
32.2 
32.2 
32.1 
32.1 
32.1 
32.1 
32.1 
32.0 
32.0 
32.0 
32.0 
32.0 
31.9 
31.9 
31.9 
31.9 
31.9 
31.8 
31.8 
31.8 
31.8 
31.8 
31.7 
31.7 
31.7 
31.7 
31.7 
31.6 
31.6 



31 

31 

31 

31 

31 

31 

31 

31 

31 

31.4 

31.4 

31.4 

31.4 

31.3 

31.3 

31.3 

31.3 

31.3 

31.2 

31.2 



Cosine. 



D. 10 



.923591 
923509 
923427 
923345 
923263 
923181 
923098 
923016 
922933 
922851 
922768 

.922686 
922603 
922520 
922438 
922355 
922272 
922189 
922106 
922023 
921940 

.921857 
921774 
921691 
921607 
921524 
921441 
921357 
921274 
921190 
921107 

.921023 
920939 
920856 
920772 
920688 
920604 
920520 
920436 
920352 
920268 

.920184 
920099 
920015 
919931 
919846 
919762 
919677 
919593 
919508 
919424 

.919339 
919254 
919169 
919085 
919000 
918915 
918830 
918745 
918659 
918574 



13.7 

13.7 

13 

13.7 

13.7 

13.7 

13 

13 

13 

13 

13 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.2 

14.2 

14.2 

14.2 



Tang. 



9.812517 
812794 
813070 
813347 
813623 
813899 
814175 
814452 
814728 
815004 
815279 

9.815555 
815831 
816107 
816382 
816658 
816933 
817209 
817484 
817759 
818035 

9.818310 
818585 
818860 
819135 
819410 
819684 
819959 
820234 
820508 
820783 
821057 
821332 
821606 
821880 
822154 
822429 
822703 
822977 
823250 
823524 

9.823798 
824072 
824345 
824619 
824893 
825166 
825439 
825713 
825986 
826259 
.826532 
826805 
827078 
827351 
827624 
827897 
828170 
828442 
828715 
828987 
Cotaug. 



D. 10' 



46.1 
46.1 
46.1 
46.0 
46.0 
46.0 
46.0 
4&.0 
46.0 
46.0 
46.0 
45.9 
45.9 
45.9 
45.9 
45.9 
45.9 
45.9 
45.9 
45.9 
45.8 
45.8 
45.8 
45.8 
45.8 
45.8 
45.8 
45.8 
45.8 
45.7 
45.7 
45.7 
45.7 
45.7 
45.7 
45.7 
45.7 
45.7 
45.6 
45.6 
45.6 
45.6 
45.6 
45.6 
45.6 



Cotang. 



45.6 
45.6 
45.5 
45.5 
45.5 
45.5 
45.5 
45.5 
45.5 



10.187482 
187206 
186930 
186653 
186377 
186101 
185825 
185548 
185272 
184996 
184721 

10.184445 
184169 
183893 
183618 
183342 
183067 
182791 
182516 
182241 
181965 

10.181690 
181415 
181140 
180865 
180590 
180316 
180041 
179766 
179492 
179217 

10.178943 
178668 
178394 
178120 
177846 
177571 
177297 
177023 
176750 
176476 

10.176202 
175928 
175655 
175381 
175107 
174834 
174561 
174287 
174014 
173741 

10.173468 
173195 
172922 
172649 
172376 
172103 
171830 
171558 
171285 
171013 
Tang. 



N. sine. N. cos. 



54464 
54488 
54513 
54537 
54561 
54586 
54610 
54635 
54659 
54683 
54708 
54732 
54756 
54781 
54805 
54829 
54854 



83851 
83835 
83819 
83804 
83788 
83772 
83756 
83740 
83724 
08 
83692 
83676 
83660 
83645 
83629 
83613 



54878)83597 

54902 

54927 

54951 

54975 

54999 

55024 

55048 

55072 

55097 

55121 

55145 

55169 



83581 
83565 
83549 
83533 
83517 
83501 
83485 
83469 
83453 
83437 
83421 
83405 

55194J83389 

5521883373 

55242 83356 

55266 

55291 

55315 

55339 

55363 

55388 

55412 

55436 

55460 

55484 

55509 

55533 

55557 

55581 

55605 



83340 
83324 
83308 
83292 
83276 
83260 
83244 
83228 
83212 
83195 
83179 
83163 
83147 
83131 
83115 
5563083098 
55654 83082 
55678J83066 
55702 83050 



55726 
55750 
55775 
55799 

55823 
55847 
55871 
55895 



83034 
83017 
83001 

82985 
82969 
82953 
82936 

82920 



55919(82904 

N. cos.|N.sine. 



TABLE II. 



Log. Sines and Tangents. (34°) Natural Sines. 



55 



9 
10 
II 
12 

13 
14 
15 

16 
17 
18 

19 
20 
21 
22 
23 
24 
25 
26 
27 
23 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 

9.747562 
747749 
747936 
748123 
748310 
748497 
748683 
748870 
749056 
749243 
749426 

9.749615 
749801 
749987 



31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 

750172 | 3 
750358 ^ 
750543 I *" 
750729 ' dU 



I). 10" 



750914 
751099 
751284 

9.751469 
751654 
751839 
752023 
752208 
752392 
752576 
752760 
752944 
753128 

9.753312 
753495 
753679 
753862 
754046 
754229 j 



754412 ! "X 
754595 ! ta 
754778 j S 
754960 ! on 



9.755143 
755326 
755508 
755690 



755872 :*"• 

756054 1 %7l 
756236 ;*X * 
756418 ! ™' 

756600 ion 
756782 i^* 

.756963 ion 
757144 oT 

757326 m 



757507 
757688 
757869 
758050 
758230 
758411 
758591 
Cosine. I 



Cosine. 

9.918574 
9184S9 
918404 
918318 
918233 
918147 
918062 
917976 
917891 
917805 
917719 

9.917634 
917548 
917462 
917376 
917290 
917204 
917118 
917032 
916946 
916859 

9.916773 
916687 
916600 
916514 
916427 
916341 
916254 
916167 
916081 
915994 

9.915907 
915820 
915733 
915646 
915559 
915472 
915385 
915297 
915210 
915123 
915035 
914948 
914860 
914773 
914685 
914598 
914510 
914422 
914334 
914246 
914158 
914070 
913982 
913894 
913806 
913718 
913630 
913541 
913453 
913365 
Sine! 



D. 10" Tang. 



14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14.3 

14.3 

14 

14 

14 

14 

14 

14.4 

14.4 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14, 

14 

14, 

14.5 

14.6 

14.5 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 



14 

14 

14 

14 

14 

14 

14.7 

14.7 

14.7 

14.7 



829260 
829532 
829805 
830077 
830349 
830621 
830893 
831165 
831437 
831709 
1.831981 
832253 
832525 
832796 
833068 
833339 
833611 
833882 
834154 
834425 
.834696 
834967 
835238 
835509 
835780 
836051 
836322 
836593 
836864 
837134 
.837405 
837675 
837946 
838216 
838487 
838757 
839027 
839297 
839568 
839838 
.840108 
840378 
840647 
840917 
841187 
841457 
841726 
841996 
842266 
842535 
.842805 
843074 
843343 
843612 
12 



D. 10' 



844151 
844420 
844689 
844958 
845227 
Cotaner. 



45.4 

45.4 

45.4 

45.4 

45.4 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.1 

45.1 

45.1 

45.1 

45.1 

45.1 

45.1 

45.1 

45.1 

45.1 

45.0 

45.0 

45.0 

45.0 

45.0 

45.0 

45.0 

45.0 

45.0 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.8 

44.8 

44.8 

44.8 

44.8 



10. 



Cotang. ! N .sine 

10. 171013 ll 55919 

170740 |j 55943 

170468i55968 

170195 

169923 

169651 

169379 

169107 

168835 

168563 

168291 
10.168019 

167747 ! 

167475 

167204 

166932 

166661 

166389 

166118 

165846 

165575 

165304 

165033 

164762 

164491 

164220 

163949 

163678 

163407 

163136 
162866 
10.162595 
162325 
162054 
161784 
161513 
161243 
160973 
160703 
160432 
160162 
10.159892 
159622 
159353 
159083 
158813 
158543 
158274 
158004 
157734 
157465 
10.157195 
156926 
156657 
156388 
156118 
155849 
155580 
155311 
155042 
154773 



T."iu 



\ 55992 
56016 
56040 
56064 
56088 
56112 
56136 
56160 
56184 
56208 
56232 
56256 
56280 
56305 
56329 
56353 
56377 
56401 
56425 
56449 
56473 
56497 
56521 
56545 
56569 
56593 
56617 
56641 
56665 
56689 
56713 
56736 
56760 
| 56784 
1 56808 
568S2 
56856 
56880 
56904 
56928 
56952 
56976 
57000 
57024 
57047 
57071 
57095 
57119 
57143 
57167 
57191 
57215 
57238 
57262 
572S6 
57310 
57334 
5735b 



N. cos. I 

32904 
82887 
82871 
82855 
82839 
82822 



82790 

82773 

82757 

82741 

82724 

82708 

82692 

82675 

82659 

82643 

82626 

82610 

82593 

82577 

82561 

82544 

82528 

82511 

82495 

82478 

82462 

82446 

82429 

82413 

82396 

82380 

82363 

82347 

82330 

82314 

82297 

82281 

82264 

82248 

82231 

82214 

82198 

82181116 

821651 15 



82148 
82132 
82116 
82098 
82082 
82065 
82048 
82032 
82015 
81999 
81982 
81965 
81949 
81932 
81915 
X. cos. Vsine. 



55 Degrees. 



JO 



Log. Sines and Tangents. (35°) Natural Sines. 



TABLE II. 



B ine. 

.75*591 
758772 
758962 

759132 
759312 
759492 
759672 
759852 
760031 
760211 
760390 
9.700569 
76074S 
760927 
761106 
761285 
761464 
761642 
761821 
761999 
762177 

.762356 
762534 
762712 
762889 
763067 
763245 
763422 
763600 
763777 
763954 

.764131 
764308 
704485 
764662 
764838 
765015 
765191 
765367 
765544 
765720 

.765896 
766072 
766247 
766423 
766598 
766774 
766949 
767124 
767300 
767475 

.767649 
767824 
767999 
768173 
768348 
768522 
768697 
768871 
769045 
769219 

Cosine. 



30.1 
30.0 
30.0 
30.0 
30.0 
30.0 
29.9 
29.9 
29.9 
129.9 
129.9 
J29.8 
129.8 
29.8 
J29.8 
29.8 
29.8 
29.7 
29 
99 

2!) 

29 
29 
29 

•J!) 
2!) 
29 
29 

■J!) 

29 
29 
29 
29 

29 
29 
29 
29 
29, 
29 
29 
29 

■J!) 
2!) 
29 
29 
29 
29 
29 
29 
2!) 
29 

99 

29 
29 
29 
2!) 
■J9 
29 
29 
29 



9.913365 
913276 
913187 
913099 
913010 
912922 
912833 
912744 
912655 
912566 
912477 

9.912388 
912299 
912210 
912121 
912031 
911942 
911853 
911763 
911674 
911584 
.911495 
911405 
911315 
911226 
911136 
911046 
910956 
910866 
910776 
910686 
910596 
910506 
910415 
910325 
910236 
910144 
910054 
909963 
909873 
909782 
909691 
909601 
909510 
909419 
909328 
909237 
909146 
909055 
908964 
908873 
908781 
908690 
90S5y9 
908507 
908416 
908324 
908233 
908141 
908049 
907958 



Sine. 



D. 10 



4.8 

4.S 

4.8 

4.S 
4.9 

4 . 9 
4.9 

4 . 9 
4.9 

4 . 9 
4.9 

1.9 
4.9 

4.9 
4.9 
5.0 
5.0 
5 . 
5.0 
5.0 
5 . 

5.6 

5.0 
5.0 
5.0 
5.0 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 



5.2 
5.2 

5 . 2 
5.2 
5.2 
6.2 
5.2 
5.2 
5 . -2 
5.3 
5.3 
r.,3 

5 . ;i 
5.3 



_laiig 1 _ 

1.845227 
845496 
845764 
846033 
846302 
846570 
846839 
847107 
847376 
847644 
847913 

1.848181 
848449 
848717 
848986 
849254 
849622 
849790 
850058 
850325 
850593 

1.850861 
851129 
851396 
851664 
851931 
852199 
852466 
852733 
853001 
853268 

'.853535 
853802 
854069 
854336 
854603 
854870 
855137 
855404 
855671 
855938 

1.856204 
856471 
856737 
857004 
857270 
857537 
857803 
858069 
858336 
858602 

1.858868 
859134 
859400 
H59000 
859932 
860198 
860464 
860730 
860995 
.scijoi 

Cot&ng. 



D. 10' 

44.8 

44. 8 

44.8 

44.8 

44.8 

41.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.5 

44.6 

44.5 

44.5 

44.5 

44.6 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44.4 

44.4 

44.4 

44.4 

44.4 

44.4 



Coteag. N. puio.lN. cob 



10, 



10 



10 



44 

44 

41 

41 

44 

44 

44.3 

44.3 

44.3 

44.3 

44.3 



10 



10 



10 



154773 
154504 
154236 
153967 
153698 
153430 
153161 
152893 
152624 
152356 
152087 
151819 
151551 
151283 
151014 
150746 
150478 
150210 
149942 
149675 
149407 
149139 
148871 
14S604 
148336 
148069 
147801 
147534 
147267 
146999 
146732 
146465 
146198 
145931 
145664 
145397 
145130 
144863 
144596 
144329 
144062 
143796 
143529 
143263 
142996 
142730 
142463 
142197 
141931 
141664 
141398 
141132 
140866 
140600 
140334 
140068 
139802 
139536 
139270 
139005 
138739 



67358 SUM 5 
57381 ~ 
57405 81882 



67429B1866 

67453 M-,- 
57477:81832 
57501 B1816 
57524B1798 
6764881782 
57572181765 
57696 181748 
57619 SI 731 
57643S1714 
57667 81698 
57691 81681 
57716B1664 
67738(81647 
57762 81631 
57786S1614 
5781081597 
57833 81580 
57857 81563 
5788181546 
67904 81530 
57928 81513 
57952 81496 
57976,81479 
67999 81462 
58023 81445 
58047 81428 
58070 81412 
58094 81395 
68118 81378 
68141J81361 
6816581344 
68189 81327 
5821J 81310 
58236 81293 
58260 81276 
58283 81259 
68307.81242 
58330 8 1JJ5 
58351 SI -20S 
58378J81191 
58401 SI 174 
58426 81157 
58449 81140 
5S472|81123 
58496 SI 106 
68619181089 
6854381072 
58567 ]S 1055 
68690B1038 
68614B1021 
68637 si 004 
5866180987 
68684480970 
68708 80953 



68731 
68766 



58779 Mi9ir. 



Tang. | N. oof, \ -•n, 



S0J36 
80919 



TABLE II. 



Log. Sines and Tangents. (36°) Natural Sines. 



57 



Sine. 
9.769-219 

; 

769566 

769740 

769913 
770087 
770260 
770433 
770606 
770779 
770952 
771125 
771298 
771470 
771643 
771815 
771987 
772159 
772331 
772503 
772675 

9.772847 
773018 
773190 
773361 
773533 
773704 
773875 
774046 
774217 
774388 

9.774558 
774729 
774899 
775070 
775240 
775410 
775580 
775750 
775920 
776090 
776259 
776429 
776598 
776768 
776937 
777106 
777275 
777444 
777613 
777781 
77950 
778119 
778287 
778455 
778624 
778792 
778960 
779128 
779295 
779463 



D. lo 



0.71 



Cosine. 



29.0 
28. 9 

28.9 

28.9 
28.9 
38 . 8 
2S.8 
28.8 
28.8 
28.8 
28.8 



28.7 
28.7 
28.7 
28.7 
28.7 
23.7 
28.6 
28.6 



28 

28 

28 

28.4 

28.3 

28 

28 



3 
3 

3 
3 
3 
2 
2 
2 

28.2 
28.2 
28.1 
28.1 
28.1 
28.1 
28.1 
28.1 
28.0 
28.0 
28.0 
28.0 
28.0 
28.0 
27.9 



Cosine. 

.907958 
907866 

907774 
907689 

907590 
907498 
907406 

907314 
907222 
907129 
907037 

1.906945 
906852 
906760 
906667 
906575 
906482 
906389 
906296 
906204 
906111 

.906018 
905925 
905832 
905739 
905645 
905552 
905459 
905366 
905272 
905179 

.905085 
904992 
904898 
904804 
904711 
904617 
904523 
904429 
904335 
904241 

.904147 
904053 
903959 
903864 
903770 
903676 
903581 
903487 
903392 
903298 

.903202 
903108 
903014 
902919 
902824 
902729 
902634 
902539 
902444 
902349 



D. lo" 



15.3 
15.3 
15.3 
15.3 
15.3 
15.3 
15.3 
15.4 
15.4 
15.4 
15.4 
15.4 
15.4 
15.4 
15.4 
15.4 
15.4 
15.5 
15.5 
15.5 
15.5 
15.5 
15.5 
15.5 
15.5 
15.5 
15.5 
15.5 
15.6 
15.6 
15.6 
15.6 
15.6 
15.6 
15.6 
15.6 
15.6 
15.6 



15 

15 

J 5 

15 

15 

15 

15 

15 

15 

15 

15.7 

15.8 

15.8 

15.8 

15.8 

15.8 

15.8 

15.8 

15.8 

15.8 

15.9 

15.9 



9.86126] 
B61621 

861792 
862058 
862323 
862589 
862854 
863119 
863385 
863650 
863915 
9.864180 
864445 
864710 
864975 
865240 
865505 
865770 
866035 
866300 
866564 
9.866829 
867094 
867358 
867623 
867887 
868152 
868416 
868680 
868945 
869209 
9.869473 
869737 
870001 
870265 
870529 
870793 
871057 
871321 
871585 
871849 
5.872112 
872376 
872640 
872903 
873167 
873430 
873694 
873957 
874220 
874484 
3.874747 
875010 
875273 
875536 
875800 
876063 
876326 
876589 
876851 
877114 
Cotans. 



1). 10 



44.3 
44.3 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 



44.0 
44.0 
44.0 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.8 
43.8 
43.8 
43.8 
43.8 
43.8 
43.8 



Ootang. N.i 



10.138739 
13S473 
138208 
137942 
137677 
137411 
137146 
136881 
136615 
136350 
136085 

10.135820 
135555 
135290 
135025 
134760 
134495 
134230 
133965 
133700 
133436 

10.133171 
132906 
132642 
132377 
132113 
131848 
131584 
131320 
131055 
130791 

10.130527 
130263 
129999 
129735 
129471 
129207 
128943 
128679 
128415 
128151 

10.127888 
127624 
127360 
127097 
126833 
126570 
126306 
126043 
125780 
125510 

10.125253 
124990 
124727 
124464 
124200 
123937 
123674 
123411 
123149 
122886 
Tiinx. 



in.. V 0O8. 



5877980902 

58802 - 
58826 80807 
6884980860 

58873 80833 
58896 80816 
58920 80799 
5894880782 
58967 81 17 66 
68990 80748 
69014 80730 
69037 8U7 13 
59061 80696 
59084 80U79 
59108 80662 
59131 S0644 
59154 80027 
59178 80610 
69201 80593 
59225 80576 
69248 80558 
5927280541 
59295 80524 
5931880507 
59342,80489 
59365 80472 
59389 80455 
59412 80438 
59436 80422 
59459 80403 
59482 80386 
59506 80368 
59529 80351 
59552,80334 
59576'80316 
59599 80299 
69622 '80282 
59646 80264 
59669 ; 80247 
59693 80230 
59716 80212 
59739 80195 
59763180178 
59786 80160 
59809 80143 
59832 80125 
5985680108 
59879,80091 
59902 S0073 
59926 180056 
59949 S0038 
59972180021 
59995 80003 
60019|79986 
60042|79968 
6006579951 
60089 79934 



60112 
60135 
60158 
60182 



N. cos. N. sine 



79916 
79899 
79881 

79864 



53 Degrees 



58 



Log. Sines and Tangents. (37°) Natural Sines. 



TABLE II. 



' Sine. D. 10" Cosine. |D. 10 



Tang. 



9.877114 

877377 
877640 
877903 
878165 
878428 
878691 
878953 
879216 
879478 
879741 

9.880003 
880265 
880528 
880790 
881052 
881314 
881576 
881839 
882101 
882363 

9.882625 
882887 
883148 
883410 
883672 
883934 
884196 
884457 
884719 
884980 
885242 
885503 
885765 
886026 
886288 
886549 
886810 
887072 
887333 
887594 

9.887855 
16 
77 
888639 
888900 
889160 
889421 
889682 
889943 
890204 
1.890465 
890725 
890986 
891247 
891507 
891768 
892028 
892289 
892549 
892810 
Co tana;. 
52 Deu 



D. 10" 



43.8 
43.8 
43.8 
43.8 
43.8 
43.8 
43.8 



Cotang. 



10 



43 

43 

43 

43 

43 

43 

43 

43 

43 

43.7 

43.7 

43.7 

43.7 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43 

43 

43 

43 

43 

43 

43 

43 

43 

43 

43 

43.5 

43.5 

43.5 

43.5 

43.5 

43.5 

43.5 

43.5 

43.5 

43 

43 

43 

43 

43 

43 

43 

43.4 

43.4 

43.4 

43.4 



10 



10 



10 



10. 



10 



. 122886 
122623 
122360 
122097 
121835 
121572 
121309 
121047 
120784 
120522 
120259 
119997 
119735 
119472 
119210 
118948 
118686 
118424 
118161 
117899 
117637 
117375 
117113 
116852 
116590 
116328 
116066 
115804 
115543 
115281 
115020 
114758 
114497 
114235 
113974 
113712 
113451 
113190 
112928 
112667 
112406 
112145 
111884 
111623 
111351 
111100 
110840 
110579 
110318 
110057 
109796 
109535 
109275 
109014 
108753 
108493 ! 
108232 ; 
107972 i 
107711 | 
107451 ! 



N.sine. N. cos 



60182 
60205 
60228 
60251 
60274 
60298 
60321 
60344 
60367 
60390 
60414 
60437 
60460 
60483 
60506 
60529 
60553 
60576 
60599 
60622 
60645 
60668 
60891 
60714 
60738 
60761 
60784 
60807 
60830 
60853 
60876 
60899 
I 60922 
60945 
60968 
60991 
61015 
61038 
I 61061 
61084 
16110 
61130 
61153 
61176 
61199 
61222 
61245 
61268 



79864 
79846 
79829 
79811 
79793 
79776 
79758 
79741 
79723 
79706 
79688 
79671 
79658 
79635 
79618 
79600 
79583 
79565 
79547 
79530 
79512 
79494 
79477 
79459 
79441 
79424 
79406 
79388 
79371 
79353 
79335 
79318 
79300 
79282 
79264 
79247 
79229 
79211 
79193 

9176 
79158 
79140 
79122 
79105 
7908? 
79069 

9051 
79033 



61291 79016 
61314 78998 
6133778980 
61360178962 
61383J78944 
61408|?b926 
61429 7o908 



! 61451 
| 61474 

61497 
i 61520 

61543 
107190 jj 6156b 



Tang, i! N. cos. N.sine 



78891 
78873 
78855 
78837 
78819 
78801 



TABLE n. 



Log. BincB and Tangents. 



Natural SineB. 



59 



Sine. 



D. 10" 



9.789342 
789504 
789665 
789827 
789988 
790149 
790310 
790471 
790632 
790793 
790954 
.791115 
791275 
791436 
791596 
791757 
791917 
792077 
792237 
792397 
792557 

9.792716 
792876 
793035 
793195 
793354 
793514 
793673 
793832 
793991 
794150 

9.794308 
794467 
794626 
794784 
794942 
795101 
795259 
795417 
795575 
795733 

9.795891 
796049 
796206 
796364 
796521 
796679 
796836 
796993 
797150 
797307 

9.797464 
797621 
797777 
797934 
798091 
798247 
798403 
798560 
798716 
798872 
Cosine. 



26.9 
26.9 
26.9 
26.9 
26.9 
26.9 
26.8 
26.8 
26.8 
26.8 



26.7 
26.7 
26.7 
26.7 
26.7 
26.7 
26.6 
26.6 
26.6 
26.6 
26.6 
26.6 
26.5 
26.5 
26.5 
26.5 
26.5 
26.5 
26.4 
26.4 



26 
26 
26 
26 
26 
26 
26 
26 

26-3 

1 26-3 

126.3 

126.3 

I26-2 

26.2 

1 26-2 

,26-2 

26-2 

26-1 

261 

26-1 

26-1 

26-1 

26-1 

26-1 

26.1 

26.0 

26.0 

26.0 



Cosine. 



D. 10" 



.896532 
896433 
896335 
896236 
896137 
896038 



895840 
895741 
895641 
895542 
.895443 
895343 
895244 
895145 
895045 
894945 
894846 
894746 
894646 
894546 
.894446 
894346 
894246 
894146 
894046 
893946 
893846 
893745 
893645 
893544 
.893444 
893343 
893243 
893142 
893041 
892940 
892839 
892739 
892638 
892536 
.892435 
892334 
892233 
892132 
892030 
891929 
891827 
891726 
891624 
891523 
1.891421 
891319 
891217 
891115 
891013 
890911 
890809 
890707 
890605 
890503 
~Sine. 



16.4 
16.5 
16.5 



16 

16 

16 

16 

16 

16 

16.5 

16.5 

16.6 

16.6 

16.6 

16.6 

16.6 

16.6 

16.6 

16.6 

16.6 

16.6 

16 

16 

16 

16 

16 

16 

16 

16 

16.7 

16.7 

16.8 

16.8 

16.8 

16.8 

16.8 

16.8 

16.8 

16.8 

16.8 

16.8 

16.9 

16.9 

16.9 

16.9 

16.9 

16.9 

16.9 

16.9 

16.9 

17.0 

17.0 

17.0 

17.0 

17.0 

17.0 

17.0 

17.0 

17.0 

17.0 



Tang. 



D. 10" 



892810 
893070 
893331 
893591 
893851 
894111 
894371 
894632 
894892 
895162 
895412 
895672 
895932 
896192 
896452 
896712 
896971 
897231 
897491 
897751 
898010 
898270 
898530 



899049 
899308 
899568 
899827 
900086 
900346 
900605 
.900864 
901124 
901383 
901642 
901901 
902160 
902419 
902679 
902938 
903197 
.903455 
903714 
903973 
904232 
904491 
904750 
905008 
905267 
905526 
905784 
.906043 
906302 
906560 
906819 
907077 
907336 
907594 
907852 
908111 
908369 
Cotang. 



43.4 
43.4 



43.4 

43.4 

43.4 

43.4 

43.4 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43 

43 

43 

43 

43 

43 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.0 



Cotang. [N. sine. N. cos 



10, 



10. 



10 



10 



10 



10 



1 Q7190 i 
106930 
106669 
106409 
106149 
105889 
105629 
105368 
105108 
104848 
104588 
104328 
104068 
103808 
103548 
103288 
103029 
102769 
102509 
102249 
101990 
101730 
101470 
101211 
100951 
100692 
100432 
100173 
099914 
099654 
099395 
099136 
098876 
098617 
098358 
098099 
097840 
097581 
097321 
097062 
096803 
096545 
096286 
096027 
095768 
095509 
095250 
094992 
094733 
094474 
094216 
093957 
093698 
093440 
093181 
092923 
092664 
092406 
092148 
091889 
091631 



61566 
61589 
61612 
61635 
61658 
61681 
61704 
61726 
61749 
61772 
61795 
61818 
61841 
61864 
61887 
61909 
61932 
61955 
61978 
62001 
1 62024 
62046 
62069 
62092 
62115 
62138 
62160 
62183 
62206 
62229 
62251 
62274 
62297 
62320 
62342 
62365 
62388 
62411 
62433 
62456 
62479 
62502 
62524 
62647 
62670 
62592 
62615 
62638 
62660 
62683 
62706 
62728 
62751 
62774 
62796 
62819 



Tang. 



62864 
62887 
62909 
62932 



78801 
78783 
78765 
78747 
78729 
78711 
78694 
78676 
78658 
78640 
78622 
78604 
78586 
78568 
78550 
78532 
78514 
78496 
78478 
78460 
78442 
78424 
78405 
78387 
78369 
7S351 
78333 
78315 
78297 
78279 
78261 
78243 
78226 
78206 
78188 
78170 
78152 
78134 
78116 
78098 
78079 
78061 
78043 
78025 
78007 
77988 
77970 
77952 
77934 
77916 
77897 
77879 
77861 
77843 
77824 
77806 



62842 77788 



77769 
77751 
77733 
77716 



N. cos. N.sine 



51 Deereos. 



20 



60 



Log. Sines and Tangents. (39°) Natural Sines. 



Sine. 



9.798772 
799028 
799184 
799339 
799495 
799651 
799806 
799962 
800117 
800272 
800427 

9.800582 
800737 



801047 
801201 
801356 
801511 
801665 
801819 
801973 
.802128 
802282 
802436 
802589 
802743 
802897 
803050 
803204 
803357 
803511 
.803664 
803817 
803970 
804123 
804276 
804428 
804581 
804734 
804886 
805039 
.805191 
805343 
805495 
805647 
805799 
805951 
806103 
806254 
806406 
806557 
.806709 
806860 
807011 
807163 
807314 
807465 
807615 
807766 
807917 
808067 



Cosine. 



D. 10" 



26.0 
26.0 
26.0 
25.9 
25.9 
25.9 
25.9 
25.9 
25.9 
25.8 
25.8 
25.8 
25.8 
25.8 
25.8 
25.8 
25.7 
25.7 
25.7 
25.7 
25.7 
25.7 
25.6 
25.6 
25.6 
25.6 
25.6 
25.6 
25.6 
25.5 
25.5 
25.5 
25.5 
25-5 
25.5 
25-4 
25.4 
25.4 
25.4 
25.4 
25.4 
25.4 
25-3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.2 
25.2 
25.2 
25.2 
25.2 
25.2 
25.2 
25.1 
25.1 
25.1 
25.1 



Cosine. 



9.890503 
890400 
890298 
890195 
890093 



889785 
889682 
889579 
889477 
.889374 
889271 
889168 
889064 



888755 
888651 
888548 
888444 

9.888341 
888237 
888134 
888030 
887926 
887822 
887718 
887614 
887510 
887406 

9.887302 
887198 
887093 
886989 
886885 
886780 
886676 
886571 



9.886257 
886152 
886047 
885942 
885837 
885732 
885627 
885522 
885416 
885311 

9.885205 
885100 
884994 
884889 
884783 
884677 
884572 
884466 
884360 
884254 



Sine. 



D. 10' 



17.0 
17.1 
17.1 
17.1 
17.1 
17.1 
17.1 
17.1 
17.1 
17.1 
17.1 
17.2 
17.2 
17.2 
17.2 
17.2 
17.2 
17.2 
17.2 
17.2 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.4 
17.4 
17.4 
17.4 



17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17.5 

17.5 

17.5 

17.5 

17.5 

17.5 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 



T ang. 

,908369 
908628 



909144 
909402 
909660 
909918 
910177 
910435 
910693 
910951 

.911209 
911467 
911724 
911982 
912240 
912498 
912756 
913014 
913271 
913529 

.913787 
914044 
914302 
914560 
914817 
915076 
915332 
915590 
915847 
916104 

.916362 
916619 
916877 
917134 
917391 
917648 
917905 
918163 
918420 
918677 

.918934 
919191 
919448 
919705 
919962 
920219 
920476 
920733 
920990 
921247 

.921503 
921760 
922017 
922274 
922530 
922787 
923044 
923300 
923557 
923813 



Cotang. 



D. 10' 



43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
43.0 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.9 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.8 
42.7 



Cotang. 



10.091631 
091372 
09.1114 
090856 
090598 
090340 
090082 
089823 
089565 
089307 
089049 

10.088791 
088533 
088276 
088018 
087760 
087502 
087244 



086729 

086471 

10-086213 

085956 



085440 
085183 
084925 
084668 
084410 
084153 
083896 

10-083638 
083381 
083123 
082866 
082609 
082352 
082095 
081837 
081580 
081323 

10-081066! 
080809 I 
080552 
080295 
080038 
079781 
079524 
079267 
079010 
078753 

10-078497 
078240 
077983 
077726 
077470 
077213 
076956 
076700 
076443 
076187 



Tang. 



N. sine. N. cos 



62932 
62955 
62977 
63000 
63022 
63045 
63068 
63090 
63113 
63135 
63158 
93180 
63203 
63225 
63248 
63271 
63293 
63316 
63338 
63361 
63383 
63406 
63428 
63451 
63473 
63496 
63518 
63540 
63563 
63585 
63608 
63630 
63653 
63675 
63698 
63720 
63742 
63765 
63787 
63810 
63832 
63854 
63877 
63899 
63922 
63944 
63966 
63989 
64011 
64033 
64056 
64078 
64100 
64123 
64145 
64167 
64190 
64212 
64234 
64256 
64279 



77715 
77696 
77678 
77660 
77641 
77623 
77605 
77586 
77568 
77550 
77531 
77513 
77494 
77476 
77458 
77439 
77421 
77402 
77384 
77366 
77347 
77329 
77310 
77292 
77273 
77255 
77236 
77218 
77199 
77181 
77162 
77144 
77125 
77107 
77088 
77070 
77051 
77033 
77014 
76996 
76977 
76959 
76940 
76921 
76903 
76884 
76866 
76847 
76828 
76810 
76791 
76772 
76754 
76735 
76717 
76698 
76679 
76661 
76642 
76623 
76604 



N. cos. N.sine, 



50 Degrees. 




Natural Sines. 



61 



| Cotang. 


N .sine. 


N. cos. 


10.076187 


64279 


76604 


075930 


64301 


76586 


075673 


64323 


76567 


075417 


64346 


76548 


075160 


64368 


76530 


074904 


64390 


76511 


074648 


64412 


76492 


074391 


64435 


76473 


074135 


64457 


76455 


073878 


64479 


76436 


073622 


64501 


76417 


10.073366 


64524 


76398 


073110 


64546 


76380 


072863 


64568 


76361 


072597 


64590 


76342 


072341 


64612 


76323 


072085 


64635 


76304 


071829 


64657 


76286 


071573 


64679 


76267 


071317 


64701 


76248 


071060 


64723 


76229 


10.070804 


64746 


76210 


070548 


64768 


76192 


070292 


64790 


76173 


070036 


64812 


76164 


069780 


64834 


76135 


069525 


64856 


76116 


069269 


64878 


76097 


069013 


64901 


76078 


068757 


64923 


76059 


068501 


64945 


76041 


10.058245 


64967 


76022 


067990 


64989 


76003 


067734 


65011 


75984 


067478 


65033 


75965 


067222 


65055 


75946 


066967 i 


65077 


75927 


066711 


66100 


75908 


066455 


65122 


75889 


066200 


65144 


75870 


065944 


65166 


75851 


10.065689 


66188 


75832 


065433 


65210 


76813 


065177 


65232 


75794 


064922 


65254 


76775 


064667 


65276 


75756 


064411 


65298 


76738 


064156 


65320 


75719 


063900 


65342 


75700 


063645 


65364 


75680 


063390 


65386 


75661 


10.063134 


65408 


76642 


062879 


65430 


76623 


062624 | 


65452 


75604 


062368 


65474 


75585 


062113 


66496 


75566 


061858 


65518 


75547 


061602 


65540 


75528 


061347 


66562 


75509 


061092 


65584 


75490 


060837 


66606 


75471 


Tang. 1 ! N. cos. 


N.aine. 



62 



Log. Sines and Tangents. (41°) Natural Sines. 



TABLE II. 



9.816943 
817088 
817233 
817379 
817524 
817668 
817813 
817958 
818103 
818247 
818392 

9.818536 
818681 
818825 
818969 
819113 
819257 
819401 
819645 
819689 
819832 

9.819976 
820120 
820263 
820405 
820550 
820693 
820836 
820979 
821122 
821265 

9.821407 
821550 
821693 
821835 
821977 
822120 
822262 
822404 
822546 
822688 

9.822830 
822972 
823114 
823255 
823397 
823539 



823821 
823963 
824104 
51 9.824245 



824386 
824527 
824668 
824808 
824949 
825090 
825230 
825371 
825511 
Cosine. 



24.2 
24.2 
24.2 
24.2 
24.1 
24.1 
24.1 
24.1 
24.1 
24.1 
24.1 
24.0 
24.0 
24.0 
24.0 
24.0 
24.0 
24.0 
23.9 
23.9 
23.9 
23.9 
23.9 
23.9 
23.9 
23.8 
23.8 
23.8 
23.8 
23.8 
23.8 
23.8 
23.8 
23.7 
23.7 
23.7 
23.7 
23.7 
23.7 
23.7 
23.6 
23.6 
23.6 
23.6 
23.6 
23.6 
23.6 
23.5 
23.5 
23.6 
23.6 
23.5 
23.6 
23.5 
23.4 
23.4 
23.4 
23.4 
23.4 
23.4 



D. 10" Co sine. 

9.877780 
877670 
877560 
877450 
877340 
877230 
877120 
877010 
876899 
876789 
876678 

9.876568 
876457 
876347 
876236 
876125 
876014 
875904 
875793 
875682 
876571 

9.875459 
875348 
875237 
875126 
875014 
874903 
874791 
874680 
874568 
874456 
874344 
874232 
874121 
874009 
873896 
873784 
873672 
873560 
873448 
873335 

9.873223 
873110 
872998 
872885 
872772 
872659 
872547 
872434 
872321 
872208 

9.872095 
871981 
871868 
871755 
871641 
871528 
871414 
871301 
871187 
871073 



D. 10' 



Sine. 



18.3 

18.3 

18.3 

18.3 

18.3 

18.4 

18.4 

18 

18 

18 

18 

18 

18 

18 

18.5 

18.5 

18.5 

18.5 

18.5 

18.5 

18.5 

18.5 

18.5 

18.5 

18.6 

18.6 

18.6 

18.6 

18.6 

18.6 

18.6 

18.6 

18.7 

18.7 

18.7 

18.7 

18.7 

18.7 

18.7 

18.7 

18.7 

18.7 

18.8 

18.8 

18.8 

18.8 

18.8 

18.8 

18.8 

18.8 

18.8 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 



Tang. 



9.939163 
939418 
939673 
939928 
940183 
940438 
940694 
940949 
941204 
941458 
941714 

9.941968 
942223 
942478 
942733 
942988 
943243 
943498 
943752 
944007 
944262 

9.944517 
944771 
945026 
945281 
945535 
945790 
946045 
946299 
946554 
946808 

9.947063 
947318 
947572 
947826 
948081 



948590 
948844 
949099 
949353 
949607 
949862 
950116 
950370 
950625 
950879 
951133 
951388 
951642 
951896 
9.952150 
952405 
952659 
952913 
953167 
953421 
953675 
953929 
954183 
954437 



D. 10" 



42.5 

42.5 

42.5 

42.5 

42.6 

42.5 

42.5 

42,5 

42.5 

42.5 

42.6 

42.5 

42.5 

42.5 

42.5 

42.5 

42.5 

42.5 

42.5 

42.5 

42.5 

42.5 

42.4 

42.4 

42.4 

42 

42 



Cotang. I |N. sine. N. cos. 



42 

42 

42 

42 

42 

42 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 



42 

42 

42 

42 

42 

42 

42 

42.4 

42.4 

42.4 

42.4 

42.3 

42.3 

42.3 

42.3 

42.3 



Cotang. 



10.060837 
060582 
060327 
060072 
059817 
059562 
059306 
059051 
058796 
058542 
058286 

10.058032 
057777 
057522 
057267 
057012 
056757 
056502 
056248 
055993 
055738 

10.055483 
055229 
054974 
054719 
054465 
054210 
053955 
053701 
053446 
053192 

10.052937 
052682 
052428 
052174 
051919 
051664 
051410 
051156 
050901 
050647 

10.050393 
050138 
049884 
049630 
049375 
049121 
048867 
048612 



65606 

65628 

65650 

65672 

65694 

65716 

65738 

65759 

65781 

65803 

65825 

65847 

65869 

65891 

65913 

65935 

65956 

65978 

66000 

66022 

66044 

66066 

66088 

66109 

66131 

66153 

66176 

66197 

66218 

66240 

66262 

66284 

66306 

66327 

66349 

66371 

66393 

66414 

66436 

66458 

66480 

66501 

66523 

66545 

66566 

66588 

66610 

66632 

66653 



0483581! 66675 
048104 j 66697 
10.047850 |,66718 
047595 | j 66740 
047341 1 66762 
047087 1 66783 
046833 166805 
046579 166827 
046326 166848 
046071 j 66870 
045817 66891 
045563 j! 66913 
Tang. i| N. cos. 



75471 
75452 
75433 
76414 
75395 
75375 
75356 
75337 
75318 
75299 
75280 
75261 
75241 
75222 
75203 
75184 
75165 
76146 
5126 
75107 
75088 
75069 
75050 
75030 
75011 
74992 
74973 
74953 
74934 
74916 
74896 
74876 
74857 
74838 
74818 
74799 
74780 
74760 
74741 
74722 
74703 
74683 
74663 
74644 
74625 
74606 
4586 
74567 
74548 
74522 
74509 
4489 
74470 
74451 
74431 
74412 
74392 
74373 
74353 
74334 
74314 



N.sinc 



48 Degrees 



TABLE n. 



Log. Sines and Tangents. (42°)* Natural Sines. 



63 



Sine. 



D. 10" 



.825511 
825651 
825791 
825931 
826071 
826211 
826351 
826491 
826631 
826770 
826910 

9.827049 
827189 
827328 
827467 
827606 
827745 
827884 
828023 
828162 
828301 

9.828439 
828578 
828716 
828855 
828993 
829131 
829269 
829407 
829545 
829683 

9.829821 
829959 
830097 
830234 
830372 
830509 
830646 
830784 
830921 
831058 

9.831195 
831332 
831469 
831606 
831742 
831879 
832015 
832152 
832288 
832425 

9 832561 
832697 
832833 
832969 
833105 
833241 
833377 
833512 
833648 
833783 



23.4 
23.3 
23.3 
23. 3 
23.3 
23.3 
23.3 
23.3 
23.3 
23.2 
23.2 



Cosine. 



23 

23 

•23 

23 

23 

23 

23 

23 

23.1 

23.1 

23.1 

23.1 

23.1 

23.0 

23.0 

23.0 

23.0 

23.0 

23.0 

23.0 

22.9 

22.9 

22.9 

22.9 

22.9 

22.9 

22.9 

22.9 

22.8 

22.8 

22.8 

22.8 

22.8 

22.8 

22.8 

22.8 

22.7 

22.7 

22.7 

22.7 

22.7 

22.7 

22.7 

22.6 

22.6 

22.6 

22.6 

22.6 

22.6 



871073 
870960 
870846 
870732 
870618 
870504 
870390 
870276 
870161 
870047 
869933 
9.869818 
869704 
869589 
869474 
869360 
869245 
869130 
869015 
868900 
868785 
868670 
868555 
868440 
868324 
868209 
868093 
867978 
867862 
867747 
867631 
867515 
867399 
867283 
867167 
867051 
866935 
866819 
866703 
866586 
866470 
866353 
866237 
866120 
866004 
865887 
865770 
865653 
865536 
865419 
865302 
865185 
865068 
864950 
864833 
864716 
864598 
864481 
864363 
864245 
864127 



D. 10" 



Cosine. 



bine. 



19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.4 
19.4 
19.4 
19.4 
19.4 
19.4 
19.4 
19.4 
19.5 
19.5 
19.5 
19.5 
19.5 
19.5 
19.5 
19.6 
19.5 
19.5 
19.6 
19.6 
19.6 
19.6 
19.6 
19.6 



Tang. 



D. 10" 



.954437 
954691 
954945 
955200 
955454 
955707 
955961 
956215 
956469 
956723 
956977 

1.957231 
957485 
957739 
957993 
958246 
958500 
958754 
959008 
959262 
959516 

). 959769 
960023 
960277 
960531 
960784 
961038 
961291 
961545 
961799 
962052 

1.962306 
962560 
962813 
963067 
963320 
963574 
963827 
964081 
964335 
964588 

1.964842 
965095 
965349 
965602 
965855 
966109 
966362 
966616 
966869 
967123 

1.967376 
967629 
967883 
968136 
968389 
968643 
968896 
969149 
969403 
969656 



42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42 

42 

42 

42 

42 

42 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 



Cotang. 

47 Degrees. 



Cotang. 



10.045563 
045309 
045056 
044800 
044546 
044293 
044039 
043785 
043531 
043277 
043023 

10.042769 
042515 
042261 
042007 
041754 
041500 
041246 
040992 
040738 
040484 

10.040231 
039977 
039723 
039469 
039216 
038962 
038709 
038455 
038201 
037948 

10.037694 
037440 
037187 
036933 
036680 
036426 
036173 
035919 
035665 
035412 

10.035158 
034905 
034651 
034398 
034145 
033891 
033638 
033384 
033131 
032877 

10.032624 
032371 
032117 
031864 
031611 
031357 
031104 
030851 
030597 
030344 



N. sine. N. cos 



Tang. 



66913 
66935 
66956 
66978 
66999 
67021 
67043 
67064 
67086 
67107 
67129 
67151 
67172 
67194 
67215 
67237 
67258 
67280 
67301 
67323 
67344 
67366 
67387 
67409 
67430 
67452 
67473 
67495 
67516 
67538 
67559 
67580 
67602 
67623 
67645 
67666 
67688 
67709 
67730 
67752 
67773 
67795 
67816 
67837 
67859 
67880 
67901 
67923 
67944 
67965 
67987 
68008 
68029 
68051 
68072 
68093 
68115 
68136 
68157 
68179 
68200 



74314 
74295 
74276 
74256 
74237 
74217 
74198 
74178 
74159 
74139 
74120 
74100 
74080 
74061 
74041 
74022 
74002 



73963 
73944 
73924 
73904 

3885 
73865 
73846 
73826 
73806 
73787 
73767 
73747 
73728 
73708 
73688 
73669 
73649 
73629 
73610 
73590 
73570 
73551 
73531 

3511 
73491 
73472 
73452 
73432 
73413 
73393 
73373 
73353 
73333 
73314 
73294 
73274 
73254 
73234 
73215 
73195 
73175 
73155 
73135 
N. coi<. N.sine. 



Log. Sines and Tangents. (44°) Natural Sines. 



65 





1 

2 
3 

4 
5 
6 

7 

8 

9 

10 

11 
12 

13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
2b 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
3? 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
/53 
54 
55 
56 
57 
58 
59 
60 



Sine. 

9.841771 
841902 
842033 
842163 
842294 
842424 
842555 
842685 
842815 
842946 
843076 

9.843206 
843336 
843466 
843595 
843725 
843855 
843984 
844114 
844243 
844372 

9.844502 
844631 
844760 
844889 
845018 
845147 
845276 
845405 
845533 
845662 

9.845790 
845919 
846047 
846175 
846304 
846432 
846560 
846688 
846816 
846944 

9.847071 
847199 
847327 
847454 
847582 
847709 
847836 
847964 
848091 
848218 

9.848345 
848472 
848599 
848726 
848852 
848979 
849106 
849232 
849359 
849485 



D. 10" 



21.8 
21.8 
21.8 
21.7 



21 

21 

21 

21 

21 

21 

21 

21.6 

21.6 

21.6 

21.6 

21.6 

21.6 

21.6 

21.5 

21.5 



21 

21 

21 

21 

21 

21 

21 

21 

21 

21 

21 

21 

21 

21.4 

21.4 

21.4 

21.3 

21.3 

21.3 

21.3 

21.3 

21.3 

21.3 

21.3 

21.2 

21.2 

21.2 

21.2 

21.2 

21.2 

21.2 

21.2 

21.1 

21.1 

21.1 

21.1 

21.1 

21.1 

21.1 

21.1 



Cosine. 

9.856934 
856812 
856690 
856568 
856446 
856323 
856201 
856078 
855956 
856833 
855711 

9.855588 
855465 
855342 
855219 
855096 
854973 
854850 
854727 
854603 
854480 

9.854356 
854233 
854109 



853862 
853738 
853614 
853490 
853366 
853242 
853118 
852994 
852869 
852745 
852620 
852496 
852371 
852247 
852122 
851997 
851872 
851747 
851622 
851497 
851372 
851246 
851121 
850996 
850870 
850745 
9.850619 
850493 
850368 
850242 
850116 
849990 
849864 
849738 
849611 
849485 



D. 10' 



20.3 
20.3 
20.4 
20.4 
20.4 
20.4 



20 

20 

20 

20 

20 

20 

20 

20 

20 

20 

20 

20 

20 

20 

20.6 

20.6 

20.6 

20.6 

20.6 

20.6 

20.6 

20.7 

20.7 

20.7 

20.7 

20.7 

20.7 

20.7 

20.7 

20.7 

20.8 

20.8 

20.8 

20.8 

20.8 

20.8 

20.8 

20.8 

20.9 

20.9 

20.9 

20.9 

20.9 

20.9 

20.9 

20.9 

21.0 

21.0 

21.0 

21.0 

21.0 

21.0 

21.0 

21.0 



Tang. 

9.984837 
985090 
985343 
985596 



986101 
986354 
986607 
986860 
987112 
987365 

9.987618 
987871 
988123 
988376 
988629 
988882 
989134 
989387 
989640 
989893 

9.990145 
990398 
990651 
990903 
991156 
991409 
991662 
991914 
992167 
992420 

9.992672 
992925 
993178 
993430 
993683 
993936 
994189 
994441 
994694 
994947 

9.995199 
995452 
995705 
995957 
996210 
996463 
996715 
996968 
997221 
997473 

3.997726 
997979 
998231 
998484 
998737 
998989 
999242 
999495 
999748 

10.000000 



Co tang. 



D. 10' 

42.1 
42.1 
42.1 
42.1 
42.1 
42.1 



42.1 



42 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 



42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 
42.1 



Cotang. 



10.015163 
014910 
014657 
014404 
014152 
013899 
013646 
013393 
013140 
012888 
012635 

10.012382 
012129 
011877 
011624 
011371 
011118 
010866 
010613 
010360 
010107 

10.009855 
009602 
009349 
009097 
008844 
008591 
008338 
008086 
007833 
007580 

10-007328 
007075 
006822 
006570 
006317 
006064 
005811 
005559 
005306 
005053 

10-004801 
004648 
004295 
004043 
003790 
003537 
003285 
003032 
002779 
002527 

10-002274 
002021 
001769 
001516 
001263 
001011 
000758 
000505 
000253 
000000 



N. sine 



69466 
69487 
69508 
69529 
69549 
69570 
69591 
69612 
69633 
69654 
69675 



Tan« 



69717 
69737 
69758 
69779 
69800 
69821 
69842 
69862 
69883 
69904 
69925 
69946 
69966 
69987 
70008 
70029 
70049 
70070 
70091 
70112 
70132 
70153 
70174 
70195 
70215 
70236 
70257 
70277 
70298 
70319 
70339 
70360 
70381 
70401 
70422 
70443 
70463 
70484 
70505 
70525 
70546 
70567 
70587 
70608 
70628 
70649 
70670 
70690 
70711 



j N. cos, 



71934 
71914 
71894 
71873 
71853 
71833 
71813 
71792 
71772 
71752 
71732 
71711 
71691 
71671 
71660 
71630 
71610 
71590 
71569 
71649 
71529 
71508 
71488 
71468 
71447 
71427 
71407 
71386 
71366 
71345 
71325 
71305 
71284 
71264 
71243 
71223 
71203 
71182 
71162 
71141 
71121 
71100 
71080 
71059 
71039 
71019 
70998 
70978 
70957 
70937 
70916 
70896 
70875 
70855 
70834 
70813 
70793 
70772 
70752 
70731 
70711 



45 Degrees. 



66 



LO GARITHMS 



TABLE III. 

LOGARITHMS OF NUMBERS. 

From 1 to 200, 

INCLUDING TWELVE DECIMAL PLACES 



jsr. 

i 

2 
3 
4 
6 

6 

7 

8 

9 

10 

11 

12 
13 
14 
15 

16 
17 
18 
19 

20 

21 

22 
23 
24 
25 

26 
27 
28 
29 
30 

31 
32 
33 

34 
35 



37 

38 
39 
40 



Log. 



000000 000000 
301029 995664 
477121 254720 
602059 991328 
698970 004336 

778151 250384 
845098 040014 
903089 986992 
954242 509439 
Same as to 1. 

041392 685158 
079181 246048 
113943 352307 
146128 035678 
176091 259056 

204119 982656 
230448 921378 
255272 505103 
278753 600953 
Same as to 2. 

322219 2947 
342422 680822 
361727 836018 
380211 241712 
397940 008672 

414973 347971 
431363 764159 
447158 031342 
462397 997899 
Same as to 3. 

491361 693834 
505149 978320 
518513 939878 
531478 917042 
544068 044350 

556302 500767 
568201 724067 
579783 596617 
591064 607026 
Same as to 4. 



1ST. 


Log. 


41 


612783 856720 


42 


623249 290398 


43 


633468 455580 


44 


643452 676486 


45 


653212 513775 


46 


662757 831682 


47 


672097 857926 


48 


681241 237376 


49 


690196 080028 


50 


Same as to 5. 


51 


707570 176098 


52 


716003 343635 


53 


724275 869601 


54 


732393 769823 


55 


740362 689494 


56 


748188 027006 


57 


755874 855672 


58 


763427 993563 


59 


770852 011642 


60 


Same as to 6. 


61 


785329 835011 


62 


792391 699498 


63 


799340 549453 


64 


806179 973984 


65 


812913 356643 


66 


819543 935542 


67 


826074 802701 


68 


832508 912706 


69 


838849 090737 


70 


Same as to 7. 


71 


851258 348719 


72 


857332 496431 


73 


863322 860120 


74 


869231 719731 


75 


875061 263392 


76 


880813 592281 


77 


886490 725172 


78 


892094 602690 


79 


897627 091290 


80 


Same as to 8. 



N. 


Log. 


81 


908485 018879 


82 


913813 852384 


83 


919078 092376 


84 


924279 286062 


85 


929418 925714 


86 


934498 451244 


87 


939519 252619 


88 


944482 672150 


89 


949390 006645 


90 


Same as to 9. 


91 


959041 392321 


92 


963787 827346 


93 


968482 948554 


94 


973127 853600 


95 


977723 605889 


96 


982271 233040 


97 


986771 734266 


98 


991226 075692 


99 


995635 194598 


100 


Same as to 10, 


101 


004321 373783 


102 


008600 171762 


103 


012837 224705 


104 


017033 339299 


105 


021189 299070 


106 


025305 865265 


107 


029383 777685 


108 


033423 755487 


109 


037426 497941 


110 


game as to 11. 


111 


045322 978787 


112 


049218 022670 


113 


053078 443483 


114 


056904 851336 


115 


060697 840354 


116 


064457 989227 


117 


068185 861746 


118 


071882 007306 


119 


075546 961393 


120 


Same as to 12. 



OF NUMBERS 



67 



N. 


Log. 


1 *' 


Log. 


B". 


Log 


121 


082785 370316 


148 


170261 715395 


175 


243038 048686 


122 


086359 830675 


149 


173186 268412 


176 


245512 667814 


123 


089905 111439 


150 


176091 259056 


177 


247973 266362 


124 


093421 685162 


151 


178976 947293 


178 


250420 002309 


125 


096910 013008 


152 


181843 587945 


179 


252853 030980 


126 


100370 545118 


153 


184691 430818 


180 


255272 505103 


127 


103803 720956 


154 


187520 720836 


181 


257678 574869 


128 


107209 969648 


155 


190331 698170 


182 


260071 387985 


129 


110589 710299 


156 


193124 588354 


183 


262451 089730 


130 


Same as to 13. 


157 


195899 652409 


184 


264817 823010 


131 


117271 295656 


158 


198657 086954 


185 


267171 728403 


132 


120573 931206 


159 


201397 124320 


186 


269512 944218 


133 


123851 640967 


160 


204119 982656 


187 


271841 606536 


134 


127104 798365 


161 


206825 876032 


188 


274157 849264 


135 


130333 768495 


162 


209515 014543 


189 


276461 804173 


136 


133538 908370 


163 


212187 604404 


190 


278753 600953 


137 


136720 567156 


164 


214843 848048 


191 


281033 367248 


138 


139879 086401 


165 


217483 944214 


192 


283301 228704 


139 


143014 800254 


166 


220108 088040 


193 


285557 309008 


140 


146128 035678 


167 


222716 471148 


194 


287801 729930 


141 


149219 112655 


168 


225309 281726 


195 


290034 611362 


142 


152288 344383 


169 


227886 704614 


196 


292256 071356 


143 


155336 037465 


170 


230448 921378 


197 


294466 226162 


144 


158362 492095 


171 


232996 110392 


198 


296665 190262 


145 


161368 002235 


172 


235528 446908 


199 


298853 076410 


146 


164352 855784 


173 


238046 103129 






147 


167317 334748 ; 


174 


240549 248283 







LOGARITHMS OF THE PRIME NUMBERS 

From 200 to 1543, 

INCLUDING TWELVE DECIMAL PLACES. 



N. 
201 
203 
207 
209 
211 



227 
229 
233 
239 

241 
251 
257 
263 
269 

271 



Log. 


BT. 


303196 057420 


277 


307496 037913 


281 


315970 345457 


283 


320146 286111 


293 


324282 455298 


307 


348304 863048 


311 


356025 857193 


313 


359835 482340 


317 


367355 921026 


331 


378397 900948 


337 


382017 042575 


347 


399673 721481 


349 


409933 123331 


353 


419955 748490 


359 


429752 280002 


367 


432969 290874 


373 



Log. 


BP. 


442479 769064 


379 


448706 319905 


383 


451786 435524 


389 


466867 620354 


397 


487138 375477 


401 


492760 389027 


409 


495544 337546 


419 


501059 262218 


421 


519827 993776 


431 


527629 900871 


433 


540329 474791 


439 


542825 426959 


443 


547774 705388 


449 


555094 448578 


457 


564666 064252 


461 


571708 831809 


463 



Log. 



578639 209968 
583198 773968 
589949 601326 
598790 506763 
603144 372620 

611723 308007 
622214 022966 
624282 095836 
634477 270161 
636487 896353 

642424 520242 
646403 726223 
652246 341003 
659916 200070 
663700 925390 

665580 991018 





68 




LOGARITHMS 






N. 


Log. 


N. 


Log. 


jsr. 

"1171 


Log. 




467 


Gjyj.lo b8U§t>6 


821 


914343 167119 


068556 895072 




479 


680335 513414 


823 


915399 835212 


1181 


072249 807613 




487 


687^28 961215 


827 


917505 509553 


1187 


074450 718955 




491 


691081 482123 


829 


918554 530550 


1193 


076640 443670 




499 


698100 545623 


839 


923761 960829 


1201 


079543 007385 




503 


701537 985056 


853 


930949 031168 


1213 


083860 800845 




609 


706717 782337 


857 


932980 821923 


1217 


085290 578210 




521 


716837 723300 


859 


933993 163831. 


1223 


087426 458017 




523 


718501 688867 


863 


936010 795715 


1229 


089551 882866 




541 


733197 265107 


877 


942999 593356 


1231 


090258 052912 




547 


737987 326333 


881 


944975 908412 


1237 


092369 699609 




557 


745855 195174 


883 


945960 703578 


1249 


096562 438356 




563 


750508 394851 


887 


947923 619832 


1259 


100025 729204 




569 


755112 266395 


907 


957607 287080 


1277 


106190 896808 




571 


756636 108246 


911 


959518 376973 


1279 


106870 542460 




577 


761175 813156 


919 


963315 511386 


1283 


108226 656362 




587 


768638 101248 


929 


968015 713994 


1289 


110252 917337 




593 


773054 693364 


937 


971739 590888 


1291 


110926 242517 




599 


777426 822389 


941 


973589 623427 


1297 


112939 986066 




601 


778874 472002 


947 


976349 979003 


1301 


114277 296540 




607 


783138 691075 


953 


979092 900638 


1303 


114944 415712 




613 


787460 474518 


967 


985426 474083 


1307 


116275 587564 




617 


790285 164033 


971 


987219 229908 


1319 


120244 795568 




619 


791690 649020 


977 


989894 563719 


1321 


120902 817604 




631 


800029 359244 


983 


992553 517832 


1327 


122870 922849 




641 


806858 029519 


991 


996073 654485 


1361 


133858 125188 




643 


808210 972924 


997 


998695 158312 


1367 


135768 514554 




647 


810904 280669 


1009 


003891 166237 


1373 


137670 537223 




653 


814913 181275 


1013 


005609 445360 


1381 


140193 678544 




659 


818885 414594 


1019 


008174 184006 


1399 


145817 714122 




661 


810201 459486 


1021 


009025 742087 


1409 


148910 994096 




673 


828015 064224 


1031 


013258 665284 


1423 


153204 896557 




677 


830588 668685 


1033 


014100 321520 


1427 


154424 012366 




683 


834420 703682 


1039 


016615 547557 


1429 


156032 228774 




691 


839478 047374 


1049 


020775 488194 


1433 


156246 402184 




701 


845718 017967 


1051 


021602 716028 


1439 


158060 793919 




709 


850646 235183 


1061 


025715 383901 


1447 


160468 531109 




719 


856728 890383 


1063 


026533 264523 


1451 


161667 412427 




727 


861534 410859 


1069 


028977 705209 


1453 


162265 614286 




733 


865103 974742 


1087 


036229 544086 


1459 


164055 291883 




739 


868644 488395 


1091 


037824 750588 


1471 


167612 672629 




743 


870988 813761 


1093 


038620 161950 


1481 


170555 058512 




751 


855639 937004 


1097 


040206 627575 


1483 


171141 151014 




757 


879095 879500 


1103 


042595 512440 


1487 


172310 968489 




761 


881384 656771 


1109 


044931 546149 


1489 


172894 731332 




769 


885926 339801 


1117 


048053 173116 


1493 


174059 807708 




773 


888179 493918 


1123 


050379 756261 


1499 


175801 632866 




787 


895974 732359 


1129 


052693 941925 


1511 


179264 464329 




797 


901458 321396 


1151 


001075 323630 


1523 


182699 903324 




809 


907948 521612 


1153 


061829 307295 


1531 


184975 190807 




811 


909020 854211 


1163 J 


065579 714728 


1543 


188365 926053 



OF NUMBERS. 69 


AUXILIARY LOGARITHMS, 




JN. 


Log. 


isr. 


Log. 




1.009 


003891166237 i 




1.0009 


000390689248 - 








1.008 


003460532110 




1.0008 


000347296684 








1.007 


003029470554 




1.0007 


000303899784 








1.006 


002598080685 




1.0006 


000260498547 








1.005 


002166061756 


>A 


1 . 0005 


000217092970 


^R 






1.004 


001733712775 


1.0004 


000173683057 








1.003 


001300933020 




1 . 0003 


000130268804 








1.002 000867721529 




1.0002 


000086850211 






1.001 J 000434077479 J 


i 


1.0001 


000043427277 








1.00009 


Log. 


N. 


Log. 




000039083266 


1 . 000009 


000003908628 






1.00008 


000034740691 


1 . 000008 


000003474338 






1.00007 


000030398072 


1.000007 


000003040047 






1.00006 


000026055410 


1.000005 


000002605756 






1.00005 


000021712704 


1 . 000005 


000002171464 






1.00004 


000017371430 


1 . 000004 


000001737173 






1.00003 


000013028638 


1.000003 


000001302880 






1.00002 


000008685802 


1.000002 


000000868587 






1.00001 


000004342923 1.000001 


000000434294 1 




K 


Log. 






1.0000001 


000000043429 (n) 




1.00000001 


000000004343 (o) 






1.000000001 


000000000434 (p) 






1.0000000001 


000000000043 (q ) 





?! 



m=0.43429448l9 log. —1.637784298. 

By the preceding tables — and the auxiliaries A, B, and 
C, we can find the logarithm of any number, true to at least 
ten decimal places. 

But some may prefer to use the following direct formula, 
which may be found in any of the standard works on algebra: 

Log. (z+l)=log.g+0.8685889638/'_L \ 

The result will be true to twelve decimal places, if z be 
over 2000. 

The log. of composite numbers can be determined by the 
combination of logarithms, already in the table, and the prime 
numbers from the formula. 

Thus, the number 3083 is a prime number, find its loga- 
rithm. 

We first find the log. of the number 3082. By factoring, 
we discover that this is the product of 46 into 67. 



70 NUMBERS. 

Log. 46, 1.6627578316 

Log. 67, 1.8260748027 

Log. 3082 3.4888326343 

0.8685889638 



Log. 3083=3.4888326343- 



6165 



NUMBERS AND THEIR LOGARITHMS, 

OFTEN USED IN COMPUTATIONS. 

Circumference of a circle to dia. 1 1 Log. 

Surface of a sphere to diameter IV =3.14159265 0.4971499 
Area of a circle to radius 1 ) 

Area of a circle to diameter 1 = .7853982 —1.8950899 
Capacity of a sphere to diameter 1 = .5235988 —1.7189986 
Capacity of a sphere to radius 1 =4.1887902 0.6220886 

Arc of any circle equal to the radius =57°29578 1.7581226 
Arc equal to radius expressed in sec. = 206264"8 5.31 44251 
Length of a degree, (radius unity) = .01 745329 —2.2418773 

12 hours expressed in seconds, = 43200 4.6354837 

Complement of the same, =0.00002315 —5.3645163 

360 degrees expressed in seconds, = 1296000 6.1126050 

A gallon of distilled water, when the temperature is 62° 
Fahrenheit, and Barometer "30 inches, is 277. r VoV cubic 
inches. 



^277.274=16.651542 nearly. 



4 
4 



277.274 
.775398 



= 18.78925284 J 231 =15.198684. 



J 282 =16.792855. 



= 18.948708. 



.785398 

The French Metre=3.2808992, English feet linear mea- 
sure, =39.3707904 inches, the length of a pendulum vi- 
brating seconds. 









TRAVERSE 


TABLE. 




71 


1 


KDeg. 


IDeg. 


V/ % Deg. 


2 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


1. 00 


0. 


01 


1. 00 


0. 


02 


1. 


00 


0. 


03 


1. 00 


0. 03 


2 


2. 00 


0. 


02 


2. 00 


0. 


03 


2. 


00 


0. 


05 


2. 00 


0. 07 


3 


3. 00 


0. 


03 


3. 00 


0. 


05 


3. 


00 


0. 


08 


3. 00 


0. 10 


4 


4. 00 


0. 


03 


4. 00 


0. 


07 


4. 


00 


0. 


10 


4. 00 


0. 14 


5 


5. 00 


0. 


04 


5. 00 


0. 


09 


5. 


00 


0. 


13 


5. 00 


0. 17 


6 


6. 00 


0. 


05 


6. 90 


0. 


10 


6. 


00 


0. 


16 


6. 00 


0. 21 


7 


7. 00 


0. 


06 


7. 00 


0. 


12 


7. 


00 


0. 


18 


7. 00 


0. 24 


8 


8. 00 


0. 


07 


8. 00 


0. 


14 


8. 


00 


0. 


21 


7. 99 


0. 28 


9 


9. 00 


0. 


08 


9. 00 


0. 


16 


9. 


00 


o. 


24 


8. 99 


0. 31 


10 


10. 00 


0. 


09 


10. 00 


0. 


17 


10. 


00 


0. 


26 


9. 99 


0. 35 


11 


11. 00 


0. 


10 


11. 00 


0. 


19 


11. 


00 


0. 


28 


10. 99 


0. 38 


12 


12. 00 


0. 


10 


12. 00 


0. 


21 


12. 


00 


0. 


31 


11. 99 


0. 42 


13 


13. 00 


0. 


11 


13. 00 


0. 


23 


13. 


00 


0. 


34 


12. 99 


0. 45 


14 


14. 00 


0. 


12 


14. 00 


0. 


24 


14. 


00 


0. 


37 


13. 99 


0. 49 


15 


15. 00 


0. 


13 


15. 00 


0. 


26 


14. 


99 


0. 


39 


14. 99 


0. 52 


16 


16. 00 


0. 


14 


16. 00 


0. 


28 


15. 


99 


0- 


42 


15. 99 


0. 56 


17 


17. 00 


0. 


15 


17. 00 


0. 


30 


16. 


99 


0. 


45 


16. 99 


0. 59 


18 


18. 00 


0. 


16 


18. 00 


0. 


31 


17. 


99 


0. 


47 


17. 99 


0. 63 


19 


19. 00 


0. 


17 


19. 00 


0. 


33 


18. 


99 


0. 


50 


18. 99 


0. 66 


20 


20. 00 


0. 


17 


20. 00 


0. 


35 


19. 


99 


o. 


52 


19. 99 


0. 70 


21 


21. 00 


0. 


18 


21. 00 


0. 


37 


20. 


99 


0. 


55 


20. 99 


0. 73 


22 


22. 00 


0. 


19 


22. 00 


0. 


38 


21. 


99 


0. 


58 


21. 99 


0. 77 


23 


23. 00 


0. 


20 


23. 00 


0. 


40 


22. 


99 


0. 


60 


22. 99 


0- 80 


24 


24. 00 


0. 


21 


24. 00 


0. 


42 


23. 


99 


0. 


63 


23. 99 


0. 84 


25 


25. 00 


0. 


22 


25. 00 


0. 


44 


24. 


99 


0. 


65 


24. 98 


0. 87 


26 


26. 00 


0. 


23 


26. 00 


0. 


45 


25. 


99 


o. 


68 


25. 98 


0. 91 


27 


27. 00 


0. 


24 


27. 00 


0. 


47 


26. 


99 


0. 


71 


26. 98 


0. 94 


28 


28. 00 


0. 


24 


28. 00 


0. 


49 


27. 


99 


0. 


73 


27. 98 


0. 98 


29 


29. 00 


0. 


25 


29. 00 


0. 


51 


28. 


99 


o. 


76 


28. 98 


1. 01 


30 


30. 00 


0. 


26 


30. 00 


0. 


52 


29. 


99 


0. 


79 


29. 98 


1. 05 


35 


35. 00 


0. 


31 


34. 99 


0. 


61 


34. 


99 


0. 


92 


34. 98 


1. 22 


40 


40. 00 


0. 


35 


39. 99 


0. 


70 


39. 


99 


1. 


05 


39. 98 


1. 40 


45 


45. 00 


0. 


39 


44. 90 


0. 


79 


44. 


99 


1. 


18 


44. 97 


1. 57 


50 


50. 00 


0. 


44 


49. 99 


0. 


87 


49. 


98 


1. 


31 


49. 97 


1. 74 


55 


55. 00 


0. 


48 


54. 99 


0. 


96 


54. 


98 


1. 


44 


54. 97 


1. 92 


60 


60. 00 


0. 


52 


59. 90 


0. 


05 


59. 


98 


1. 


57 


59. 96 


2. 09 


65 


65. 00 


0. 


57 


64. 99 


1. 


13 


64. 


98 


1. 


70 


64. 96 


2. 27 


70 


70. 00 


0. 


61 


69. 99 


1. 


22 


69. 


98 


1. 


83 


69. 96 


2. 44 


75 


75. 00 


0. 


65 


74. 99 


1. 


31 


74. 


97 


1. 


96 


74. 95 


2. 62 


80 


80. 00 


0. 


70 


79. 99 


1. 


40 


79. 


97 


2. 


09 


79. 95 


2. 79 


85 


85. 00 


0. 


74 


84. 99 


1. 


48 


84. 


97 


2. 


23 


84. 95 


2. 97 


90 


90. 00 


0. 


79 


89. 99 


1. 


57 


89. 


97 


2. 


36 


89. 95 


3. 14 


95 


90. 00 


0. 


83 


94. 99 


1. 


66 


94. 


97 


2. 


49 


94. 94 


3. 32 


100 


100. 00 


0. 


87 


99. 98 
Dep. 


1. 


75 


99. 


97 


2. 


62 


99. 94 


3. 49 


Dep. 


Lat. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


89^ Deg. 


89 Deg. 


88^ Deg. 


88 Deg. 


1 „. 



72 




TRAVERSE 


TABLE. 




S 

55" 
I 

8 


2K Deg. 


3 Deg. 


3K Deg. 


4 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


1. 00 


0. 


04 


1. 


00 


0. 05 


1. 00 


0. 06 


1. 00 


0. 07 


2 


2. 00 


0. 


09 


2. 


00 


0. 10 


2. 00 


0. 12 


2. 00 


0. 14 


3 


3. 00 


0. 


13 


3. 


00 


0. 16 


2. 99 


0. 18 


2. 99 


0. 21 


4 


4. 00 


0. 


17 


3. 


99 


0. 21 


3. 99 


0. 24 


3. 99 


0. 28 


5 


5. 00 


0. 


22 


4. 


99 


0. 26 


4. 99 


0. 31 


4. 99 


0. 35 


6 


5. 99 


0. 


26 


5. 


99 


0. 31 


5. 99 


0. 37 


5. 99 


0. 42 


7 


6. 99 


0. 


31 


6. 


99 


0. 37 


6. 99 


0. 43 


6. 98 


0. 49 


8 


7. 99 


0. 


35 


7. 


99 


0. 42 


7. 99 


0. 49 


7. 98 


0. 56 


9 


8. 99 


0. 


39 


8. 


99 


0. 47 


8. 98 


0. 55 


8. 98 


0. 63 


10 


9. 99 


0. 


44 


9. 


99 


0. 52 


9. 98 


0. 61 


9. 98 


0. 70 


11 


10. 99 


0. 


48 


10. 


98 


0. 58 


10. 98 


0. 67 


10. 97 


0. 77 


12 


11. 99 


0. 


52 


11. 


98 


0. 63 


11. 98 


0. 73 


11. 97 


0. 84 


13 


12. 99 


0. 


57 


12. 


98 


0. 68 


12. 99 


0. 79 


12. 97 


0. 91 


14 


13. 99 


0. 


61 


13. 


98 


0. 73 


13. 97 


0. 85 


13. 97 


0. 98 


15 


14. 99 


o. 


65 


14. 


98 


0. 79 


14. 97 


0. 92 


14. 96 


1. 05 


16 


15. 99 


o. 


70 


15. 


98 


0. 84 


15. 97 


0. 98 


15. 96 


1. 12 


17 


16. 98 


o. 


74 


16. 


98 


0. 89 


16. 97 


1. 04 


16. 96 


1. 19 


18 


17. 98 


o. 


79 


17. 


98 


0. 94 


17. 97 


1. 10 


17. 96 


1. 26 


19 


18. 98 


o. 


83 


18. 


98 


0. 99 


18. 96 


1. 16 


18. 95 


1. 33 


20 


19. 98 


0. 


87 


19. 


97 


1. 05 


19. 96 


1. 22 


19. 95 


1. 40 


21 


20. 98 


0. 


92 


20. 


97 


1. 10 


20. 96 


1. 28 


20. 95 


1. 46 


22 


21. 98 


0. 


96 


21. 


97 


1. 15 


21. 96 


1. 34 


21. 95 


1. 53 


23 


22. 98 


1. 


00 


22. 


97 


1. 20 


22. 96 


1. 40 


22. 94 


1. 60 


24 


23. 98 


1. 


05 


23. 


97 


1. 26 


23. 96 


1. 47 


23. 94 


1. 67 


25 


24. 98 


1. 


09 


24. 


97 


1. 31 


24. 95 


1. 53 


24. 94 


1. 74 


26 


25. 98 


1. 


13 


25. 


96 


1. 36 


25. 95 


1. 59 


25- 94 


1. 81 


27 


26. 97 


1. 


18 


26. 


96 


1. 41 


26. 95 


1. 65 


26- 93 


1. 88 


28 


27.97 


1. 


22 


27. 


96 


1. 47 


27. 95 


1. 71 


27. 93 


1. 95 


29 


28. 97 


1. 


26 


28. 


96 


1.52 


28. 95 


1. 77 


28- 93 


2. 02 


30 


29. 97 


1. 


31 


29. 


96 


1.57 


29. 94 


1. 83 


29. 93 


2. 09 


35 


34. 97 


1. 


53 


34. 


95 


1. 83 


34. 93 


2. 14 


34- 91 


2. 44 


40 


39. 96 


1. 


75 


39. 


95 


2. 09 


39. 93 


2. 44 


39. 90 


2. 79 


45 


44. 96 


1. 


96 


44. 


94 


2. 36 


44. 92 


2. 75 


44- 89 


3. 14 


50 


49. 95 


2. 


18 


49. 


93 


2. 62 


49. 91 


3. 05 


49- 88 


3. 49 


55 


54. 95 


2. 


40 


54. 


92 


2. 88 


54. 90 


3. 36 


54- 87 


3. 84 


60 


59. 94 


2. 


62 


59. 


92 


3. 14 


59. 89 


3. 66 


59. 83 


4. 19 


65 


64. 94 


2. 


84 


64. 


91 


3. 40 


64. 88 


3. 97 


64. 84 


4. 53 


70 


69. 93 


3. 


05 


69. 


90 


3. 66 


69. 87 


4. 27 


69. 83' 


4. 88 


75 


74. 93 


3. 


27 


74. 


90 


3. 93 


74. 86 


4. 58 


74. 82 


5. 23 


80 


79. 92 


3. 


49 


79. 


89 


4. 19 


79. 85 


4. 88 


79. 81 


5. 58 


85 


84. 92 


3. 


71 


84. 


88 


4. 45 


84. 84 


5. 19 


84. 79 


5. 93 


90 


89. 91 


3. 


93 


89. 


98 


4. 71 


89. 83 


5. 49 


89. 78 


6. 28 


95 


94. 91 


4. 


14 


94. 


87 


4. 97 


94. 82 


5. 80 


94. 77 


6. 63 


100 


99. 91 


4. 


36 


99. 


86 


5.23 


99. 81 


6. 10 


99. 76 


6. 98 


Dep- 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


87^ 


Deg 




87 Deg. 


86^ Deg. 


86 Deg. 


1: - - 



TRAVERSE TABLE. 73 




2 

f 
g. 


4K Deg. 


5 Deg. 


5M Deg. 


6 Deg. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




1 


1. 00 


0. 08 


1. 00 


0. 09 


1. oo 


0. 10 


0. 99 


0. 10 




2 


1. 99 


0. 16 


1. 99 


0. 17 


1. 99 


0. 19 


1. 99 


0. 21 




3 


2. 99 


0. 24 


2. 99 


0. 26 


2. 99 


0. 29 


2. 98 


0. 31 




4 


3. 99 


0. 31 


3. 98 


0. 35 


3. 98 


0. 38 


3. 98 


0. 41 




5 


4. 98 


0. 39 


4. 98 


0. 44 


4. 98 


0. 48 


4. 97 


0. 52 




6 


5. 98 


0. 47 


5. 98 


0. 52 


5. 97 


0. 58 


5. 97 


0. 63 




7 


6. 98 


0. 55 


6. 97 


0. 61 


6. 97 


0. 67 


6. 96 


0. 73 




8 


7. 98 


0. 63 


7. 97 


0. 70 


7. 96 


0. 76 


7. 96 


0. 84 




9 


8. 97 


0. 71 


8. 97 


0. 78 


8. 96 


0. 86 


8: 95 


0. 94 




10 


9. 97 


0. 78 


9. 96 


0. 87 


9. 95 


0. 96 


9. 95 


1. 05 




11 


10. 97 


0. 86 


10. 96 


0. 96 


10. 95 


1. 05 


10. 94 


1. 15 




12 


11. 96 


0. 94 


11. 95 


1. 05 


11. 94 


1. 15 


11. 93 


1. 25 




13 


12. 96 


1. 02 


12. 95 


1. 13 


12. 94 


1. 25 


12. 93 


1. 36 




14 


13. 96 


1. 10 


13. 95 


1. 22 


13. 94 


1. 34 


13. 92 


1. 46 




15 


14. 95 


1. 18 


14. 94 


1. 31 


14. 93 


1. 44 


14. 92 


1. 57 




16 


15. 95 


1. 26 


15. 94 


1. 39 


15. 93 


1. 53 


15. 91 


1. 67 




17 


16. 95 


1. 33 


16. 94 


1. 48 


16. 92 


1. 63 


16. 91 


1. 78 




18 


17. 94 


1. 41 


17. 93 


1. 57 


17. 92 


1. 73 


17. 90 


1. 88 




19 


18. 94 


1. 49 


18. 93 


1. 66 


18. 91 


1- 82 


18. 90 


1. 99 




20 


19. 94 


1. 57 


19. 92 


1. 74 


19. 91 


1. 92 


19. 89 


2. 09 




21 


20. 94 


1. 65 


20. 92 


1. 83 


20. 90 


2. 01 


20. 88 


2. 20 




22 


21. 93 


1. 73 


21. 92 


1. 92 


21. 90 


2. 11 


21. 88 


2. 30 




23 


22. 93 


1. 80 


22. 91 


2. 00 


22. 89 


2. 20 


22. 87 


2. 40 




24 


23. 93 


1. 88 


23. 91 


2. 09 


23. 89 


2. 30 


23. 87 


2. 51 




25 


24. 92 


1. 96 


24. 90 


2. 18 


24. 88 


2. 40 


24. 86 


2. 61 




26 


25. 92 


2. 04 


25. 90 


2. 27 


25. 88 


2. 49 


25. 86 


2. 72 




27 


26. 92 


2. 12 


26. 90 


2. 35 


26. 88 


2. 59 


26. 85 


2. 82 




28 


27. 91 


2. 20 


27. 89 


2. 44 


27. 87 


2. 68 


27. 85 


2. 93 




29 


28. 91 


2. 28 


28. 89 


2. 53 


28. 87 


2. 78 


28. 84 


3. 03 




30 


29. 21 


2. 35 


29. 89 


2. 61 


29. 86 


2. 88 


29. 84 


3. 14 




35 


34. 89 


2. 75 


34. 87 


3. 05 


34. 84 


3. 35 


34. 81 


3. 66 


i 


40 


39. 88 


3. 14 


39. 85 


3. 49 


39. 82 


3. 83 


39. 78 


4. 18 




45 


44. 86 


3. 53 


44. 83 


3. 92 


44. 79 


4. 31 


44. 75 


4. 70 




50 


49. 85 


3. 92 


49. 81 


4. 36 


49. 77 


4. 79 


49. 73 


5. 23 




55 


54. 85 


4. 08 


54. 79 


4. 79 


54. 75 


5. 27 


54. 70 


5. 75 




60 


59. 84 


4. 45 


59. 77 


5. 23 


59. 72 


5. 75 


59. 67 


6. 27 




65 


64. 82 


4. 82 


64. 75 


5. 67 1 


64. 70 


6. 23 


64. 64 


6. 79 




70 


69. 81 


5. 19 


69. 73 


6. 10 


69. 68 


6. 71 


69. 62 


7. 32 




75 


74. 79 


5. 65 


74. 71 


6. 54 | 74. 65 


7. 19 


74. 59 


7. 84 




80 


79. 78 


5. 93 


79. 70 


6. 97 79. 63 


7. 67 i 


76. 56 


8. 36 




85 


84. 77 


6. 30 


84. 68. 


7.41 


84. 61 


8. 15 1 


84. 53 


8. 88 




90 


89. 75 


6. 67 


89. 66 


7. 84 


89. 59 


8. 63 I 


89. 51 


9. 41 




95 


94. 74 


7. 04 


94. 64 


8. 28 1 94. 56 


9. 11 94. 48 


9. 93 




100 


99. 73 


7. 41 


99. 62 


8. 72 


99. 54 


9. 58 || 99. 45 


10. 43 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. || Dep. 


Lat. 




85^ Deg. 


85 Deg. 


84% Deg. 84 Deg. 









F^ 




TRAVERSE TABLE. 




g 
s 

1 
p 


&A Deg. 


7 Deg. 


7K Deg. 


8 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0. 


99 


0. 11 


0. 99 


0. 12 


0. 99 


0. 13 


0. 99 


0. 14 


2 


1. 


99 


0. 23 


1.99 


0. 24 


1. 98 


0.26 


1. 98 


0. 28 


3 


2. 


98 


0. 34 


2. 98 


0. 37 


2 97 


0.39 


2. 97 


0. 42 


4 


3. 


97 


0. 45 


3. 97 


0. 49 


3. 97 


0.52 


3. 96 


0. 56 


5 


4. 


97 


0. 57 


4. 96 


0. 61 


4. 96 


0.65 


4. 95 


0. 70 


6 


5. 


96 


0. 68 


5. 96 


0. 73 


5. 95 


0.78 


5. 94 


0. 84 


7 


6. 


96 


0. 79 


6. 95 


0. 85 


6. 94 


0.91 


6. 93 


0. 97 


8 


7. 


95 


0. 91 


7. 94 


0. 97 


7. 93 


1.04 


7. 92 


1. 11 


9 


8. 


94 


1. 02 


8. 93 


1. 10 


8. 92 


1.17 


8.91 


1. 25 


10 


9. 


94 


1. 13 


9. 93 


1. 22 


9. 91 


1.31 


9. 90 


1. 39 


11 


10. 


93 


1. 25 


10. 92 


1. 34 


10. 91 


1.44 


10. 89 


1. 53 


12 


11. 


92 


1. 36 


11. 91 


1. 46 


11. 90 


1.57 


11. 88 


1. 67 


13 


12. 


92 


1.47 


12. 90 


1. 58 


12. 89 


1.70 


12. 87 


1. 81 


14 


13. 


91 


1. 59 


13. 90 


1. 71 


13. 88 


1.83 


13. 86 


1. 95 


15 


14. 


90 


1. 70 


14. 89 


1. 83 


14. 87 


1.96 


14. 85 


2. 09 


16 


15. 


90 


1.81 


15. 88 


1. 95 


15. 86 


2.09 


15. 84 


2. 23 


17 


16. 


89 


1. 92 


16. 87 


2. 07 


16. 85 


2.22 


16. 83 


2. 37 


18 


17. 


88 


2. 04 


17. 87 


2. 19 


17. 85 


2.35 


17. 82 


2. 51 


19 


18. 


88 


2. 15 


18. 86 


2. 32 


18. 84 


2.48 


18. 82 


2. 64 


20 


19. 


87 


2. 26 


19. 85 


2. 44 


19. 83 


2.61 


19. 81 


2. 78 


21 


20. 


87 


2. 38 


20. 84 


2. 56 


20. 82 


2.74 


20. 80 


2. 92 


22 


21. 


86 


2. 49 


21. 84 


2. 68 


21. 81 


2.87 


21. 79 


3. 06 


23 


22. 


85 


2. 60 


22. 83 


2. 80 


22. 80 


3.00 


22. 78 


3. 20 


24 


23. 


85 


2. 72 


23. 82 


2. 92 


23. 79 


3. 13 


23. 77 


3. 34 


25 


24. 


84 


2. 83 


24. 81 


3. 05 


24. 79 


3.26 


24. 76 


3. 48 


26 


25. 


83 


2. 94 


25. 81 


3. 17 


25. 78 


3.39 


25. 75 


3. 62 


27 


26. 


83 


3. 06 


26. 80 


3. 29 


26. 77 


3.52 


26. 74 


3. 76 


28 


27. 


82 


3. 17 


27. 79 


3. 41 


27. 76 


3.65 


27. 73 


3. 90 


29 


28. 


81 


3. 28 


28. 78 


3. 53 


28. 75 


3.79 


28. 72 


4. 04 


30 


29. 


81 


3. 40 


29. 78 


3. 66 


29. 74 


3.92 


29. 71 


4. 18 


35 


34. 


78 


3. 96 


34. 74 


4. 27 


34. 70 


4.57 


34. 66 


4. 87 


40 


39. 


74 


4. 53 


39. 70 


4. 87 


39. 66 


5.22 


39. 61 


5. 57 


45 


44. 


71 


5. 09 


44. 67 


5. 48 


44. 62 


5.87 


44. 56 


6. 26 


50 


49. 


68 


5. 66 


49. 63 


6. 09 


49. 57 


6.53 


49. 51 


6. 96 


55 


54. 


65 


6. 23 


54. 59 


6. 70 


55. 58 


6.70 


54. 46 


7. 65 


60 


59. 


61 


6. 79 


59. 55 


7. 31 


59. 55 


7. 31 


59. 42 


8. 35 


65 


64. 


58 


7. 36 


64. 52 


7. 92 


64. 52 


7.92 


64. 37 


9 05 


70 


69. 


55 


7. 92 


69. 48 


8. 53 


69. 48 


8.53 


69. 32 


9. 74 


75 


74. 


52 


8. 49 


74. 44 


9. 14 


74. 44 


9. 14 


74. 27 


10. 44 


80 


79. 


49 


9. 06 


79. 40 


9. 75 


79. 40 


9.75 


79. 22 


11. 13 


85 


84. 


45 


9. 62 


84. 37 


10. 36 


84. 37 


10. 36 


84. 17 


11. 83 


90 


89. 


42 


10. 19 


89. 33 


10. 97 


89. 33 


10.97 


89. 12 


12. 53 


95 


94. 


39 


10.75 


94. 29 


11.58 


94. 29 


11.58 


94. 08 


13. 22 


100 


99. 


36 


11.32 


99. 25 


12. 19 


99. 25 


12. 19 


99. 03 


13. 92 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


83^ Deg. 


83 Deg. 


82^ Deg. 


82 Deg. 







TRAVERSE TABLE. 


75 


g 

g 

8 


8K Deg. 


9 Deg. 


9 M »eg. 


10 


Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0. 99 


0. 15 


0. 99 


0. 16 


0. 99 


0. 17 


0. 98 


0. 17 


2 


1. 98 


0. 30 


1. 98 


0. 31 


1. 97 


0. 33 


1. 97 


0. 35 


3 


2. 97 


0. 44 


2. 96 


0. 47 


2. 96 


0. 50 


2. 95 


0. 52 


4 


3. 96 


0. 59 


3. 95 


0. 63 


3. 95 


0. 66 


3. 94 


0. 69 


5 


4. 95 


0. 74 


4. 94 


0. 78 


4. 93 


0. 83 


4. 92 


0. 87 


6 


5. 93 


0. 89 


5. 93 


0. 94 


5. 92 


0. 99 


5. 91 


1. 04 


7 


6. 92 


1. 03 


6. 91 


1. 10 


6. 90 


1. 16 


6. 89 


1. 22 


8 


7. 91 


1. 18 


7. 90 


1. 25 


7. 89 


1. 32 


7. 88 


1. 39 


9 


8. 90 


1. 33 


8. 89 


1. 41 


8. 88 


1. 49 


8. 86 


1. 56 


10 


9. 89 


1. 48 


9. 88 


1. 56 


9. 86 


1. 65 


9. 85 


1. 74 


11 


10. 88 


1. 63 


10. 86 


1. 72 


10. 85 


1. 82 


10. 83 


1. 91 


12 


11. 87 


1. 77 


11. 85 


1. 88 


11. 84 


1. 98 


11. 82 


2. 08 


13 


12. 86 


1. 92 


12. 84 


2. 03 


12. 82 


2. 15 


12. 80 


2. 26 


14 


13. 85 


2. 07 


13. 83 


2. 19 


13. 81 


2. 31 


13. 79 


2. 43 


15 


14. 84 


2. 22 


14. 82 


2. 35 


14. 79 


2. 48 


14. 77 


2. 60 


16 


15. 82 


2. 36 


15. 80 


2. 50 


15. 78 


2. 64 


15. 76 


2. 78 


17 


16. 81 


2.51 


16. 79 


2. 66 


16. 77 


2. 81 


16. 74 


2. 95 


18 


17. 80 


2. 66 


17. 78 


2. 82 


17. 75 


2. 97 


17. 73 


3. 13 


19 


18. 79 


2. 81 


18. 77 


2. 97 


18. 74 


3. 14 


18. 71 


3. 30 


20 


19. 78 


2. 96 


19. 75 


3. 13 


19. 73 


3. 30 


19. 70 


3. 47 


21 


20. 77 


3. 10 


20. 74 


3. 29 


20. 71 


3. 47 


20. 68 


3. 65 


22 


21. 76 


3. 25 


21. 73 


3. 44 


21. 70 


3. 63 


21. 67 


3. 82 


23 


22. 75 


3. 40 


22. 72 


3. 60 


22. 68 


3. 80 


22. 65 


3. 99 


24 


23. 74 


3. 55 


23. 70 


3. 75 


23. 67 


3. 96 


23. 64 


4. 17 


25 


24. 73 


3. 70 


24. 69 


3. 91 


24. 66 


4. 13 


24. 62 


4. 34 


26 


25. 71 


3. 84 


25. 68 


4. 07 


25. 64 


4. 29 


25. 61 


4. 51 


27 


26. 70 


3. 99 


26. 67 


4. 22 


26. 63 


4. 46 


26. 59 


4. 69 


28 


27. 69 


4. 14 


27. 66 


4. 38 


27. 62 


4. 62 


27. 57 


4. 86 


29 


28. 68 


4. 29 


28, 64 


4.54 


28. 60 


4. 79 


28. 56 


5. 04 


30 


29.67 


4. 43 


29.63 


4. 69 


29. 59 


4. 95 


29. 54 


5.21 


35 


34. 62 


5. 17 


34. 57 


5. 48 


34. 52 


5. 78 


34. 47 


6. 08 


40 


39. 56 


5. 91 


39.51 


6. 26 


39. 45 


6. 60 


39. 39 


6. 95 


45 


44. 51 


6. 65 


44. 45 


7. 04 


44. 38 


7. 43 


44. 32 


7. 81 


50 


49. 45 


7. 39 


49. 38 


7. 82 


49. 32 


8.25 


49. 24 


8. 68 


55 


54. 40 


8. 13 


54. 32 


8. 60 


54. 25 


9. 08 


54. 16 


9. 95 


60 


59. 34 


8. 87 


59. 26 


9. 39 


59. 18 


9. 90 


59. 09 


10. 42 


65 


64. 29 


9. 61 


64. 20 


10. 17 


64. 11 


10. 73 


64. 01 


11. 29 


70 


69.23 


10. 35 


69. 14 


10. 95 


69. 04 


11.55 


68. 94 


12. 16 


75 


74. 18 


11.09 


74. 08 


11. 73 


73. 97 


12. 38 


73.86 


13. 02 


80 


79. 12 


11. 82 


79. 02 


12. 51 


78. 90 


13.20 


78. 78 


13. 89 


85 


84.07 


12. 56 


83. 95 


13. 30 


83. 83 


14. 03 


83. 71 


14. 76 


90 


89.01 


13. 30 


88.89 


14. 08 


88. 77 


14.85 


88. 63 


15. 63 


95 


93.96 


14, 04 


93. 83 


14. 86 


93. 70 


15. 68 


93. 56 


16. 50 


100 


98. 90 


14.78 


98. 77 


15. 64 


98. 63 


16.50 


98. 48 


17. 36 


Dep. 


Lat. 


Dep. 


Lat 


Dep. 


Lat. 


Dep. 


Lat. 


81^ 


Deg. | 811 


eg- 


80^ Deg. 


80 Deg. 



76 




TRAVERSE TABLE. 




1 9 

§ 

1 


10% Deg. 


11 Deg. 


UK Deg- 


12 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


0. 


98 


0. 18 


0. 98 


0. 19 


0. 98 


0.20 


0. 98 


0. 21 


2 


1. 


97 


0. 36 


1.96 


0. 38 


1. 96 


0.40 


1. 96 


0. 42 


3 


2. 


95 


0. 55 


2. 94 


0. 57 


2. 94 


0.60 


2. 93 


0. 62 


4 


3. 


93 


0. 73 


3. 93 


0. 76 


3. 92 


0.80 


3. 91 


0. 83 


5 


4. 


92 


0. 91 


4. 91 


0. 95 


4. 90 


1.00 


4. 89 


1. 04 


6 


5. 


90 


1. 09 


5. 89 


1. 14 


5. 88 


1.20 


5. 87 


1. 25 


7 


6. 


88 


1. 28 


6. 87 


1. 34 


6. 86 


1.40 


6. 85 


1. 46 


8 


7. 


87 


1. 46 


7. 85 


1. 53 


7. 84 


1.59 


7. 83 


1. 66 


9 


8. 


85 


1. 64 


8. 83 


1. 72 


8. 82 


1.79 


8. 80 


1. 87 


10 


9. 


83 


1. 82 


9. 82 


1. 91 


9. 80 


1.99 


9. 78 


2. 08 


11 


10. 


82 


2. 00 


10. 80 


2. 10 


10. 78 


2. 19 


10. 76 


2. 29 


12 


11. 


80 


2. 19 


11. 78 


2. 29 


11. 76 


2.39 


11.74 


2. 49 


13 


12. 


78 


2. 37 


12. 76 


2. 48 


12. 74 


2.59 


12. 72 


2. 70 


14 


13. 


77 


2. 55 


13. 74 


2. 67 


13. 72 


2.79 


13. 69 


2. 91 


15 


14. 


75 


2. 73 


14. 72 


2. 87 


14. 70 


2.99 


14.67 


3. 12 


16 


15. 


73 


2. 92 


15. 71 


2. 86 


15. 68 


3.19 


15.65 


3. 33 


17 


16. 


72 


3. 10 


16. 69 


3. 05 


16. 66 


3.39 


16. 63 


3. 53 


18 


17. 


70 


3. 28 


17. 67 


3. 24 


17. 64 


3.59 


17. 61 


3. 74 


19 


18. 


68 


3. 46 


18. 65 


3. 43 


18. 62 


3.79 


18. 58 


3. 95 


20 


19. 


67 


3. 64 


19. 63 


3. 63 


19. 60 


3.99 


19. 56 


4. 16 


21 


20. 


65 


3. 83 


20. 61 


3. 82 


20. 58 


4.13 


20. 54 


4. 37 


22 


21. 


63 


4. 01 


21. 60 


4. 01 


21. 56 


4.39 


21. 52 


4. 57 


23 


22. 


61 


4. 19 


22. 58 


4. 20 


22. 54 


4.59 


22. 50 


4. 78 


24 


23. 


60 


4. 37 


23. 56 


4. 39 


23. 52 


4.78 


23. 48 


4. 99 


25 


24. 


58 


4. 56 


24. 54 


4. 58 


24. 50 


4.98 


24. 45 


5. 20 


26 


25. 


56 


4. 74 


25. 52 


4. 77 


25. 48 


5.18 


25. 43 


5. 41 


27 


26. 


55 


4. 92 


26. 50 


4. 96 


26. 46 


5.38 


26. 41 


5. 61 


28 


27. 


53 


5. 10 


27. 49 


5. 15 


27. 44 


5.58 


27. 39 


5. 82 


29 


28. 


51 


5. 28 


28. 47 


5. 34 


28. 42 


5.78 


28. 37 


6. 03 


30 


29. 


50 


5. 47 


29. 45 


5- 72 


29. 40 


5.98 


29. 34 


6. 24 


35 


34. 


41 


6. 38 


34. 36 


6. 68 


34. 30 


6.98 


34. 24 


7. 28 


40 


39. 


33 


7. 29 


39. 27 


7. 63 


39. 20 


7.97 


39. 13 


8. 32 


45 


44. 


25 


8. 20 


44. 17 


8. 59 


44. 10 


8.97 


44. 02 


9. 36 


50 


49. 


16 


9. 11 


49. 08 


9. 54 


49. 00 


9.97 


48. 91 


10. 40 


55 


54. 


08 


10. 02 


53. 99 


10. 49 


53. 90 


10.97 


53. 80 


11. 44 


60 


59. 


00 


10.93 


58. 90 


11. 45 


58. 80 


11.96 


58. 69 


12. 47 


65 


63. 


91 


11.85 


63. 81 


12. 40 


63. 70 


12.96 


63. 58 


13. 51 


70 


68. 


83 


12.76 


68. 71 


13. 36 


68. 59 


13.96 


68. 47 


14. 55 


75 


73. 


74 


13.67 


73. 62 


14. 31 


73. 49 


14.95 


73. 36 


15. 59 


80 


78. 


66 


14.58 


78. 53 


15. 26 


78. 39 


15.95 


78. 25 


16. 63 


85 


83. 


58 


15. 49 


83. 44 


16. 22 


83. 29 


16.95 


83. 14 


17. 67 


90 


88. 


49 


16. 40 


88. 35 


17. 17 


88. 19 


17.94 


88. 03 


18. 71 


95 


93. 


41 


17. 31 


93. 25 


18. 13 


93. 09 


18.94 


92. 92 


19. 75 


100 


98. 


33 


18.22 


98. 16 


19. 08 


97. 99 

Dep. 


19.94 


97. 81 


20. 79 


Dep. 


Lat. 


Dep. 


Lat. 


Lat. 


Dep. 


Lat. 




79^ 


Deg. 


79 Deg. 


78K Deg- 


78 


Deg. 














TRAVERSE TABLE. 


77 


3 

55' 

1 

8 


12K ©eg- 


13 Deg. 


13^ Deg. 


14 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0. 98 


0. 22 


0. 


97 


0. 23 


0. 


97 


0. 23 


0. 97 


0. 24 


2 


1. 95 


0. 43 


1. 


95 


0. 46 


1. 


95 


0. 47 


1. 94 


0. 48 


3 


2. 93 


0. 65 


2. 


92 


0. 67 


2. 


92 


0. 70 


2. 91 


0. 73 


4 


3. 91 


0. 87 


3. 


90 


0. 90 


3. 


89 


0. 93 


3. 88 


0. 97 


5 


4. 88 


1. 08 


4. 


87 


1. 12 


4. 


86 


1. 17 


4. 85 


1. 21 


6 


5. 86 


1. 30 


5. 


85 


1. 35 


5. 


83 


1. 40 


5. 82 


1. 45 


7 


6. 83 


1. 52 


6. 


82 


1. 57 


6. 


81 


1. 63 


6. 79 


1. 69 


8 


7. 81 


1. 73 


7. 


80 


1. 80 


7. 


78 


1. 87 


7. 76 


1. 94 


9 


8. 79 


1. 95 


8. 


77 


2. 02 


8. 


75 


2. 10 


8. 73 


2. 18 


10 


9. 76 


2. 16 


9. 


74 


2. 25 


9. 


72 


2. 33 


9 70 


2. 42 


11 


10. 74 


2. 38 


10. 


72 


2. 47 


10. 


70 


2. 57 


10. 67 


2. 66 


12 


11. 72 


2. 60 


11. 


69 


2. 70 


11. 


67 


2. 80 


11. 64 


2. 90 


13 


12. 69 


2. 81 


12. 


67 


2. 92 


12. 


64 


3. 03 


12. 61 


3. 15 


14 


13. 67 


3. 03 


13. 


64 


3. 15 


13. 


61 


3. 27 


13. 58 


3. 39 


15 


14. 64 


• 3. 25 


14. 


62 


3. 37 


14. 


59 


3. 50 


14. 55 


3. 63 


16 


15. 62 


3. 46 


15. 


59 


3. 60 


15. 


56 


3. 74 


15. 52 


3. 87 


17 


16. 60 


3. 68 


16. 


57 


3. 82 


16. 


53 


3. 97 


16. 50 


4. 11 


18 


17. 57 


3. 90 


17. 


54 


4. 05 


17. 


50 


4. 20 


17. 47 


4. 35 


19 


18. 55 


4. 11 


18. 


51 


4. 27 


18. 


48 


4. 44 


18. 44 


4. 60 


20 


19. 53 


4. 33 


19. 


49 


4. 50 


19. 


45 


4. 67 


19. 41 


4. 84 


21 


20. 50 


4. 55 


20. 


46 


4.72 


20. 


42 


4. 90 


20. 38 


5. 08 


22 


21. 48 


4. 76 


21. 


44 


4. 95 


21. 


39 


5. 14 


21. 35 


5. 32 


23 


22. 45 


4. 98 


22. 


41 


5. 17 


22. 


36 


5. 37 


22. 32 


5. 56 


24 


23. 43 


5. 19 


23. 


38 


5. 40 


23. 


34 


5. 60 


23. 29 


5. 81 


25 


24. 41 


5. 41 


24. 


36 


5. 62 


24. 


31 


5. 84 


24. 26 


6. 05 


26 


25. 38 


5. 63 


25. 


33 


5. 85 


25. 


28 


6. 07 


25. 23 


6 29 


27 


26. 36 


5. 84 


26. 


31 


6. 07 


26. 


25 


6. 30 


26. 20 


6. 53 


28 


27. 34 


6. 06 


27. 


28 


6. 30 


27. 


23 


6. 54 


27. 17 


6. 77 


29 


28. 31 


6.28 


28. 


26 


6. 52 


28. 


20 


6. 77 


28. 14 


7. 02 


30 


29. 29 


6. 49 


29. 


23 


6. 75 


29. 


17 


7. 00 


29. 11 


7. 26 


35 


34. 17 


7. 58 


34. 


10 


7. 87 


34. 


03 


8. 17 


33. 96 


8. 47 


40 


39. 05 


8. 66 


38. 


97 


9. 00 


38. 


89 


9. 34 


38. 81 


9. 68 


45 


43. 93 


9. 74 


43. 


85 


10. 12 


43. 


76 


10. 51 


43. 66 


10. 89 


50 


48. 81 


10. 82 


48. 


72 


11. 25 


48. 


62 


11. 67 


48. 51 


12. 10 


55 


53. 70 


11.90 


53. 


59 


12. 37 


53. 


48 


12. 84 


53. 37 


13. 31 


60 


58. 58 


12. 99 


58. 


46 


13. 50 


58. 


34 


14. 01 


58. 22 


14. 52 


65 


63. 46 


14. 07 


63. 


33 


14. 62 


63. 


20 


15. 17 


63. 07 


15. 72 


70 


68. 34 


15. 15 


68. 


21 


15. 75 


68. 


07 


16. 34 


67. 92 


16. 93 


75 


73. 22 


16. 23 


73. 


08 


16. 87 


72. 


93 


17. 50 


72. 77 


18. 14 


80 


78. 10 


17. 32 


77. 


95 


18. 00 


77. 


79 


18. 68 


77. 62 


19. 35 


85 


82. 99 


18. 40 


82. 


82 


19. 12 


82. 


65 


19. 84 


82. 48 


20. 56 


90 


87. 87 


19. 48 


87. 


69 


20. 25 


87. 


51 


21. 01 


87. 33 


21. 77 


95 


92. 75 


20. 56 


92. 


57 


21. 37 


92. 


38 


22. 18 


92. 18 


22. 98 


100 


97. 63 


21.64 


97. 


44 


22. 50 


97. 


24 


23. 34 


97. 03 


24. 19 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


W% Deg. 




77 1 


5eg. 




76^ 


Deg. 


76 


Deg. 





78 


TRAVERSE TABLE. 


g 

I 


14K Deg- 


15 Deg. 


15K Deg. 


16 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0. 97 


0.25 


0. 97 


0. 26 


0. 96 


0.27 


0. 96 


0. 28 


2 


1. 94 


0. 50 


1. 93 


0. 52 


1. 93 


0.53 


1. 92 


0. 55 


3 


2. 90 


0.75 


2. 90 


0. 78 


2. 89 


0.80 


2. 88 


0. 83 


4 


3. 87 


1. 00 


3. 86 


1. 04 


3. 85 


1.07 


3. 85 


1. 10 


5 


4. 84 


1.25 


4. 83 


1. 29 


4. 82 


1.34 


4. 81 


1. 38 


6 


5. 81 


1. 50 


5. 80 


1. 55 


5. 78 


1.60 


5. 77 


1.65 


7 


6. 78 


1.75 


6. 76 


1. 81 


6. 75 


1.87 


6. 73 


1. 93 


8 


7. 75 


2. 00 


7. 73 


2. 07 


7.71 


2. 14 


7. 69 


2. 21 


9 


8. 71 


2. 25 


8. 69 


2. 33 


8. 67 


2.41 


8. 65 


2. 48 


10 


9. 68 


2. 50 


9. 66 


2. 59 


9. 64 


2.67 


9. 61 


2. 76 


11 


10. 65 


2. 75 


10. 63 


2. 85 


10. 60 


2.94 


10. 57 


3. 03 


12 


11. 62 


3. 00 


11. 59 


3. 11 


11. 56 


3.21 


11. 54 


3. 31 


13 


12. 59 


3. 25 


12. 56 


3. 36 


12. 53 


3.47 


12. 50 


3. 58 


14 


13. 55 


3. 51 


13. 52 


3. 62 


13. 49 


3.74 


13. 46 


3. 86 


15 


14. 52 


3. 76 


14. 49 


3. 88 


14. 45 


4.01 


14. 42 


4. 13 


16 


15. 49 


4. 01 


15. 45 


4. 14 


15. 42 


4.28 


15. 38 


4. 41 


17 


16. 46 


4. 26 


16. 42 


4. 40 


16. 38 


4.54 


16. 34 


4. 69 


18 


17. 43 


4. 51 


17. 39 


4. 66 


17. 35 


4.81 


17. 30 


4. 96 


19 


18. 39 


4. 76 


18. 35 


4. 92 


18. 31 


5.08 


18. 26 


5. 24 


20 


19. 36 


5. 01 


19. 32 


5. 18 


19. 27 


5.34 


19. 23 


5. 51 


21 


20. 33 


5. 26 


20. 28 


5. 44 


20. 24 


5.61 


20. 19 


5. 79 


22 


21. 30 


5. 51 


21. 25 


5. 69 


21. 20 


5.88 


21. 15 


6. 06 


23 


22. 27 


5. 76 


22. 22 


5. 95 


22. 16 


6. 15 


22. 11 


6. 34 


24 


23. 24 


6. 01 


23. 18 


6.21 


23. 13 


6.41 


23. 07 


6. 62 


25 


24. 20 


6.26 


24. 15 


6. 47 


24. 09 


6.68 


24. 03 


6. 89 


26 


25. 17 


6. 51 


25. 11 


6. 73 


25. 05 


6.95 


24. 99 


7. 17 


27 


26. 14 


6. 76 


26. 08 


6. 99 


26. 02 


7.22 


25. 95 


7. 44 


28 


27. 11 


7. 01 


27. 05 


7.25 


26. 98 


7.48 


26. 92 


7. 72 


29 


28. 08 


7. 26 


28. 01 


7.51 


27.95 


7.75 


27 88 


7. 99 


30 


29. 04 


7. 51 


28. 98 


7.76 


28. 91 


8.02 


28. 84 


8.27 


35 


33. 89 


8. 76 


33. 81 


9. 06 


33. 73 


9.35 


33. 64 


9. 65 


40 


38. 73 


10. 02 


38. 64 


10. 35 


38. 55 


10.69 


38. 45 


11. 03 


45 


43. 57 


11. 27 


43. 47 


11. 65 


43. 36 


12.03 


43. 26 


12. 40 


50 


48. 41 


12. 52 


48. 30 


12. 94 


48. 18 


13.36 


48. 06 


13. 78 


55 


53.25 


13. 77 


53. 13 


14. 24 


53. 00 


14.70 


52. 87 


15. 16 


60 


58. 09 


15. 02 


57. 96 


15. 53 


57. 82 


16.03 


57. 68 


16. 54 


65 


62. 93 


16. 27 


62. 79 


16. 82 


62. 64 


17.37 


62. 48 


17. 92 


70 


67. 77 


17.53 


67. 61 


18. 12 


67.45 


18.71 


67. 29 


19. 29 


75 


72. 61 


18. 78 


72. 44 


19. 41 


72. 27 


20.04 


72. 09 


20. 67 


80 


77. 45 


20. 03 


77.27 


20. 71 


77. 09 


21.38 


76. 90 


22. 05 


85 


82.29 


21.28 


82. 10 


22. 00 


81. 91 


22.72 


81. 71 


23. 43 


90 


87. 13 


22. 53 


86. 93 


23. 29 


86. 73 


24.05 


86. 51 


24. 8.1 


95 


91. 97 


23. 79 


91.76 


24. 59 


91.54 


25.39 


91. 32 


26. 19 


100 


96. 81 


25. 04 


96. 59 


25. 88 


96. 36 


26.72 


96. 13 


27. 56 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


75% Deg. 


75 Deg. 


74% Deg. 


74 Deg. 





TRAVERSE TABLE. 79 


§ 


16K Deg. 


17 


Deg. 


17^ Deg- 


18 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0. 


96 


0. 


28 


0. 96 


0. 29 


0. 


95 


0. 


30 


0. 95 


0. 31 


2 


1. 


92 


0. 


57 


1. 91 


0. 58 


1. 


91 


0. 


60 


1. 90 


0. 62 


3 


2. 


88 


0. 


85 


2. 87 


0. 88 


2. 


86 


0. 


90 


2. 85 


0. 93 


4 


3. 


84 


1. 


14 


3. 83 


1. 17 


3. 


81 


1. 


20 


3. 80 


1. 24 


5 


4. 


79 


1. 


42 


4. 78 


1. 46 


4. 


77 


1. 


50 


4. 76 


1 55 


6 


5. 


75 


1. 


70 


5. 74 


1.75 


5. 


72 


1. 


80 


5. 71 


1. 85 


7 


6. 


71 


1. 


99 


6. 69 


2. 05 


6. 


68 


2. 


10 


6. 66 


2. 16 


8 


7. 


67 


2. 


27 


7. 65 


2. 34 


7. 


63 


2. 


41 


7. 61 


2. 47 


9 


8. 


63 


2. 


56 


8. 61 


2. 63 


8. 


58 


2. 


71 


8. 56 


2. 78 


10 


9. 


59 


2. 


84 


9. 56 


2. 92 


9. 


54 


3. 


01 


9 51 


3. 09 


11 


10. 


55 


3. 


12 


10. 52 


3. 22 


10. 


49 


3. 


31 


10. 46 


3. 40 


12 


11. 


51 


3. 


41 


11. 48 


3. 51 


11. 


44 


3. 


61 


11. 41 


3. 71 


13 


12. 


46 


3. 


69 


12. 43 


3. 80 


12. 


40 


3. 


91 


12. 36 


4. 02 


14 


13. 


42 


3. 


98 


13. 39 


4. 09 


13. 


35 


4. 


21 


13. 31 


4. 33 


15 


14. 


38 


4. 


26 


14. 34 


4. 39 


14. 


31 


4. 


51 


14. 27 


4. 64 


16 


15. 


34 


4. 


54 


15. 30 


4. 68 


15. 


26 


4. 


81 


15. 22 


4. 94 


17 


16. 


30 


4. 


83 


16. 26 


4. 97 


16. 


21 


5. 


11 


16. 17 


5. 25 


18 


17. 


26 


5. 


11 


17. 21 


5. 26 


17. 


17 


5. 


41 


17. 12 


5. 56 


19 


18. 


22 


5. 


40 


18. 17 


5. 56 


18. 


12 


5. 


71 


18. 07 


5. 87 


20 


19. 


18 


5. 


68 


19. 13 


5. 85 


19. 


07 


6. 


01 


19. 02 


6. 18 


21 


20. 


14 


5. 


96 


20. 08 


6. 14 


20. 


03 


6. 


31 


19. 97 


6- 49 


22 


21. 


09 


6. 


25 


21. 04 


6. 43 


20. 


98 


6. 


62 


20. 92 


6. 80 


23 


22. 


05 


6. 


53 


21. 99 


6. 72 


21. 


94 


6. 


92 


21. 87 


7. 11 


24 


23. 


01 


6. 


82 


22. 95 


7. 02 


22. 


89 


7. 


22 


22. 83 


7. 42 


25 


23. 


97 


7. 


10 


23. 91 


7. 30 


23. 


84 


7. 


52 


23. 78 


7. 73 


26 


24. 


93 


7. 


38 


24. 86 


7. 60 


24. 


80 


7. 


82 


24. 73 


8. 03 


27 


25. 


89 


7. 


67 


25. 82 


7. 89 


25. 


75 


8. 


12 


25. 68 


8. 34 


28 


26. 


85 


7. 


95 


26. 78 


8. 19 


26. 


70 


8. 


42 


26. 63 


8. 65 


29 


27. 


81 


8. 


24 


27. 73 


8. 48 


27. 


66 


8. 


72 


27. 58 


8. 96 


30 


28. 


76 


8. 


52 


28. 69 


8. 77 


28. 


61 


9. 


02 


28. 53 


9. 27 


35 


33. 


56 


9. 


94 


33. 47 


10. 23 


33. 


38 


10- 


52 


33. 29 


10. 82 


40 


38. 


35 


11. 


36 


38. 25 


11. 69 


38. 


15 


12. 


03 


38. 04 


12. 36 


45 


43. 


15 


12. 


78 


43. 03 


13. 16 


42. 


92 


13- 


53 


42. 80 


13. 91 


50 


47. 


94 


14. 


20 


47. 82 


14. 62 


47. 


69 


15- 


04 


47. 55 


15. 45 


55 


52. 


74 


15. 


62 


52. 60 


16. 08 


52. 


45 


16. 


54 


52. 31 


17. 00 


60 


57. 


53 


17. 


04 


57. 38 


17. 54 


57. 


22 


18. 


04 


57. 06 


18. 54 


65 


62. 


32 


18. 


46 


62. 16 


19. 00 


61. 


99 


19. 


55 


61. 82 


20. 09 


70 


67. 


12 


19. 


88 


66. 94 


20. 47 


66. 


76 


21- 


05 


66. 57 


21. 63 


75 


71. 


91 


21. 


30 


71. 72 


21. 93 


71. 


53 


22. 


55 


71. 33 


23. 18 


80 


76. 


71 


22. 


72 


76. 50 


23. 39 


76. 


30 


24- 


06 


76. 08 


24. 72 


85 


81. 


50 


24. 


14 


81. 29 


24. 85 


81. 


07 


25- 


56 


80. 84 


26. 27 


90 


86. 


29 


25. 


56 


86. 07 


26. 31 


85. 


83 


27. 


06 


85. 60 


27.81 


95 


91. 


09 


26. 


98 


90. 85 


27. 78 


90. 


60 


28- 


57 


90. 35 


29. 36 


100 


95. 


88 


28. 


40 


95. 63 


29. 24 


95. 


37 


30. 


07 


95. 11 


30. 90 

Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dop. 




73^ 


Deg. 


1 


73 1 


)eg. 


72^ Deg. 


72 Deg. 





80 


TRAVERSE TABLE. 






g 

1 
: 8 

1 


18^ Deg. 


19 Deg. 


19K Deg. 


20 Deg. 




Lat. 


Dep. 
0. 32 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




0. 95 


0. 95 


0. 33 


0. 94 


0.33 


0. 94 


0. 34 




2 


1. 90 


0. 63 


1. 89 


0. 65 


1. 89 


0.67 


1. 88 


0. 68 




3 


2. 84 


0. 95 


2. 84 


0. 98 


2. 83 


1.00 


2. 82 


1. 03 




4 


3. 79 


1. 27 


3. 78 


1. 30 


3. 77 


1.34 


3. 76 


1. 37 




5 


4. 74 


1. 59 


4. 73 


1. 63 


4. 71 


1.67 


4. 70 


1. 71 




6 


5. 69 


1. 90 


5. 67 


1. 95 


5.66 


2. 00 


5. 64 


2. 05 




7 


6. 64 


2. 22 


6. 62 


2. 28 


6. 60 


2.34 


6. 58 


2. 39 




8 


7. 59 


2. 54 


7. 56 


2. 60 


7.54 


2.67 


7. 52 


2. 74 




9 


8. 53 


2. 86 


8. 51 


2. 93 


8. 48 


3.01 


8.46 


3. 08 




10 


9. 48 


3. 17 


9. 46 


3. 26 


9. 43 


3.34 


9. 40 


3. 42 




11 


10. 43 


3. 49 


10. 40 


3. 58 


10. 37 


3.67 


10. 34 


3. 76 




12 


11. 38 


3. 81 


11. 35 


3.91 


11. 31 


4.01 


11.28 


4. 10 




13 


12. 33 


4. 12 


12. 29 


4. 23 


12. 25 


4.34 


12. 22 


4. 45 




14 


13. 28 


4. 44 


13. 24 


4. 56 


13. 20 


4.67 


13. 16 


4. 79 




15 


14. 22 


4. 76 


14. 18 


4. 88 


14. 14 


5.01 


14. 10 


5. 13 




16 


15. 17 


5. 08 


15. 13 


5. 21 


15. 08 


5.34 


15.04 


5. 47 




17 


16. 12 


5. 39 


16. 07 


5.53 


16. 02 


5.67 


15. 97 


5. 81 




18 


17. 07 


5.71 


17. 02 


5. 86 


16. 97 


6.01 


16.91 


6. 16 




19 


18. 02 


6. 03 


17. 96 


6. 19 


17. 91 


6.34 


17. 85 


6. 50 




20 


18. 97 


6. 35 


18. 91 


6. 51 


18. 85 


6.68 


18. 79 


6. 84 




21 


19. 91 


6. 66 


19. 86 


6. 84 


19. 80 


7.01 


19. 73 


7. 18 




22 


20. 86 


6. 98 


20. 80 


7. 16 


20. 74 


7.34 


20. 67 


7. 52 




23 


21. 81 


7. 30 


21. 75 


7.49 


21. 68 


7.68 


21. 61 


7. 87 




24 


22. 76 


7. 62 


22. 69 


7. 81 


22. 62 


8.01 


22. 55 


8. 21 




25 


23. 71 


7. 93 


23. 64 


8. 14 


23. 57 


8.34 


23. 49 


8. 55 




26 


24. 66 


8. 25 


24. 58 


8. 46 


24. 51 


8.68 


24. 43 


8. 89 




27 


25. 60 


8. 57 


25. 53 


8. 79 


25. 45 


9.01 


25. 37 


9. 23 




28 


26. 55 


8. 88 


26. 47 


9. 12 


26. 39 


9.34 


26. 31 


9,58 




29 


27. 50 


9. 20 


27. 42 


9. 44 


27. 34 


9.68 


27.25 


9. 92 




30 


28. 45 


9. 52 


28. 37 


9. 77 


28. 28 


10.01 


28. 19 


10. 26 




35 


33. 19 


11. 11 


33. 09 


11. 39 


32. 99 


11.68 


32. 89 


11. 97 




40 


37. 93 


12. 69 


37. 82 


13- 02 


37. 71 


13.35 


87. 59 


13. 68 




45 


42. 67 


14. 28 


42. 55 


14. 65 


42. 42 


15.02 


42. 29 


15. 39 




50 


47. 42 


15. 87 


47. 28 


16. 28 


47. 13 


16.69 


46. 98 


17. 10 




55 


52. 16 


17. 45 


52. 00 


17. 91 


51. 85 


18.36 


51. 68 


18. 81 




60 


56. 90 


19. 04 


56. 73 


19. 53 


56. 56 


20.03 


56. 38 


20. 52 




65 


61. 64 


20. 62 


61. 46 


21. 16 


61. 27 


21.70 


61. 08 


22. 23 




70 


66. 38 


22. 21 


66. 19 


22. 79 


67. 98 


23.37 


65. 78 


23. 94 




75 


71. 12 


23. 80 


70. 91 


24. 42 


70. 70 


25.04 


70. 48 


25. 65 




80 


75. 87 


25. 38 


75. 64 


26. 05 


75. 41 


26.70 


75. 18 


27. 36 




85 


80. 61 


26. 97 


80. 37 


27. 67 


80. 12 


28.37 


79. 87 


29. 07 




90 


85. 35 


28. 56 


85. 10 


29. 30 


84. 84 


30.04 


84. 57 


30. 78 




95 


90. 09 


30. 14 


89. 82 


30. 93 


89. 55 


31.71 


89. 27 


32. 49 




100 


94. 83 


31.73 


94.55 


32. 56 


94. 26 


33.38 


93. 97 


34. 20 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. j 


Lat. 


Dep. 


Lat. 




71K Deg. 


71 Deg. ) 


70% Deg. 


70 Deg. 













TRAVERSE TABLE. 


81 


s 

5' 

tr*- 

I 


20^ Deg. 


21 Deg. 


21^ Deg. 


22 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0. 


94 


0. 35 


0. 93 


0. 


36 


0. 


93 


0. 37 


0. 93 


0. 37 


2 


1. 


87 


0. 70 


1. 87 


0. 


72 


1. 


86 


0. 73 


1. 85 


0. 75 


3 


2. 


81 


1. 05 


2. 80 


1. 


08 


2. 


79 


1. 10 


2. 78 


1. 12 


4 


3. 


75 


1. 40 


3. 73 


1. 


43 


3. 


72 


1. 47 


3. 71 


1.50 


5 


4. 


68 


1. 75 


4. 67 


1. 


79 


4. 


65 


1. 83 


4. 64 


1. 87 


6 


5. 


62 


2. 10 


5. 60 


2. 


15 


5. 


58 


2. 20 


5. 56 


2. 25 


7 


6. 


56 


2. 45 


6. 54 


2. 


51 


6. 


51 


2. 57 


6. 49 


2. 62 


8 


7. 


49 


2. 80 


7. 47 


2. 


87 


7. 


44 


2. 93 


7. 42 


3. 00 


9 


8. 


43 


3. 15 


8. 40 


3. 


23 


8. 


37 


3. 30 


8. 34 


3. 37 


10 


9. 


37 


3. 50 


9. 34 


3. 


58 


9. 


30 


3. 67 


9 27 


3. 75 


11 


10. 


30 


3. 85 


10. 27 


3. 


94 


10. 


23 


4. 03 


10. 20 


4. 12 


12 


11. 


24 


4. 20 


11. 20 


4. 


30 


11. 


17 


4. 40 


11. 13 


4. 50 


13 


12. 


18 


4. 55 


12. 14 


4. 


66 


12. 


10 


4. 76 


12. 05 


4. 87 


14 


13. 


11 


4. 90 


13. 07 


5. 


02 


13. 


03 


5. 13 


12. 98 


5. 24 


15 


14. 


05 


5. 25 


14. 00 


5. 


38 


13. 


96 


5. 50 


13. 91 


5. 62 


16 


14. 


99 


5. 60 


14. 94 


5. 


73 


14. 


89 


5. 86 


14. 83 


5. 99 


17 


15. 


92 


5. 95 


15. 87 


6. 


09 


15. 


82 


6. 23 


15. 76 


6. 37 


18 


16. 


86 


6. 30 


16. 80 


6. 


45 


16. 


75 


6-60 


16. 69 


6. 74 


19 


17. 


80 


6. 65 


17. 74 


6. 


81 


17. 


68 


6. 96 


17. 62 


7. 12 


20 


18. 


73 


7. 00 


18. 67 


7. 


17 


18. 


61 


7. 33 


18. 54 


7. 49 


21 


19. 


67 


7. 35 


19. 61 


7. 


53 


19. 


54 


7. 70 


19. 47 


7. 87 


22 


20. 


61 


7. 70 


20. 54 


7. 


88 


20. 


47 


8. 06 


20. 40 


8. 24 


23 


21. 


54 


8. 05 


21. 47 


8. 


24 


21. 


40 


8. 43 


21. 33 


8. 62 


24 


22. 


48 


8. 40 


22. 41 


8. 


60 


22. 


33 


8. 80 


22. 25 


8. 99 


25 


23. 


42 


8. 76 


23. 34 


8. 


96 


23. 


26 


9. 16 


23. 18 


9. 37 


26 


24. 


35 


9. 11 


24. 27 


9. 


32 


24- 


19 


9. 53 


24. 11 


9. 74 


27 


25. 


29 


9. 46 


25. 21 


9. 


68 


25- 


12 


9- 90 


25. 03 


10. 11 


28 


26. 


23 


9. 81 


26. 14 


10. 


08 


26- 


05 


10. 26 


25. 96 


10. 49 


29 


27. 


16 


10. 16 


27. 07 


10. 


39 


26- 


98 


10. 63 


26. 89 


10. 86 


30 


28. 


10 


10. 51 


28. 01 


10. 


75 


27- 


91 


11- 00 


27. 82 


11. 24 


35 


32. 


78 


12. 26 


32. 68 


12. 


54 


| 32- 


56 


12- 83 


32. 45 


13. 11 


40 


37. 


47 


14. 01 


37. 34 


14. 


33 


37. 


22 


14- 66 


37. 09 


14. 98 


45 


42. 


15 


15. 76 


42. 01 


16. 


13 


41. 


87 


16- 49 


41. 72 


16. 86 


50 


46. 


83 


17. 51 


46. 68 


17. 


92 


46. 


52 


18. 33 


46. 36 


18. 73 


55 


51. 


52 


19. 26 


51. 35 


19. 


71 


51. 


17 


20- 16 


51. 00 


20. 60 


60 


56. 


20 


21. 01 


56. 01 


21. 


50 


55. 


83 


21- 99 


55. 63 


22. 48 


65 


60. 


88 


22. 76 


60. 68 


23. 


29 


60. 


48 


23- 82 


60. 27 


24. 35 


70 


65. 


57 


24.51 


65. 35 


25. 


09 


65. 


13 


25- 66 


64. 90 


26. 22 


75 


70. 


25 


26. 27 


70. 02 


26. 


88 


69. 


78 


27. 49 


69. 54 


28. 10 


80 


74. 


93 


28. 02 


74. 69 


28. 


67 


74. 


43 


29. 32 


74. 17 


29. 97 


85 


79. 


62 


29. 77 


79. 35 


30. 


46 


79. 


09 


31. 15 


78. 81 


31. 84 


90 


84. 


30 


31. 52 


84. 02 


32. 


25 


83. 


74 


32. 99 


83. 45 


33. 71 


95 


98. 


98 


33. 27 


88. 69 


34. 


04 


; 88. 


39 


34. 82 


1 88. 08 


35. 59 


, 100 


93. 


67 


35. 02 


93. 36 


35. 


84 


93. 


04 


36. 65 


92. 72 


37. 46 

Lat, 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


69J4 Deg. 


69] 


Deg. 




68^ Deg. 


68 


Deg. 





82 


TRAVERSE TABLE. 






S 

d 
a 
ft 

1 


22% Deg. 


23 Deg. 


23% Deg. 


24 Deg. 




Lat. 


Dep. 


Lat. 


Dep. 
0. 39 


Lat. 


Dep. 


Lat. 


Dep. 




0. 90 


0. 38 


0. 92 


0. 92 


0.40 


0. 91 


0. 41 




2 


1. 85 


0. 77 


1. 84 


0. 78 


1. 83 


0.80 


1. 83 


0. 81 




3 


2. 77 


1. 15 


2. 76 


1. 17 


2. 75 


1.20 


2. 74 


1. 22 




4 


3. 70 


1. 53 


3. 68 


1. 56 


3. 67 


1.59 


3. 65 


1. 63 




5 


4. 62 


1. 91 


4. 60 


1. 95 


4. 59 


1.99 


4. 57 


2. 03 




6 


5. 54 


2. 30 


5. 52 


2. 34 


5. 50 


2.39 


5. 48 


2. 44 




7 


6. 47 


2. 68 


6. 44 


2. 74 


6. 42 


2.79 


6. 39 


2. 85 




8 


7. 39 


3. 06 


7. 36 


3. 13 


7. 34 


3. 19 


7. 31 


3.25 




9 


8. 31 


3. 44 


8. 28 


3. 52 


8. 25 


3.59 


8.22 


3. 66 




10 


9. 24 


3. 83 


9. 20 


3. 91 


9. 17 


3.99 


9 14 


4. 07 




11 


10. 16 


4. 21 


10. 13 


4. 30 


10. 09 


4.39 


10. 05 


4. 47 




12 


11. 09 


4. 59 


11. 05 


4. 69 


11. 00 


4.78 


10. 96 


4. 88 




13 


12. 01 


4. 97 


11. 97 


5. 08 


11. 92 


5. 18 


11. 88 


5. 29 




14 


12. 93 


5. 36 


12. 89 


5. 47 


12. 84 


5.58 


12. 79 


5. 69 




15 


13. 86 


5. 74 


13. 81 


5. 86 


13. 76 


5.98 


13. 70 


6. 10 




16 


14. 78 


6. 12 


14. 73 


6. 25 


14. 67 


6. 38 


14. 62 


6. 51 




17 


15. 71 


6.51 


15. 65 


6. 64 


15. 59 


6.78 


15. 53 


6. 92 




18 


16. 63 


6. 89 


16. 57 


7. 03 


16. 51 


7.18 


16. 44 


7. 32 




19 


17. 55 


7. 27 


17. 49 


7.42 


17. 42 


7.58 


17. 36 


7. 73 




20 


18. 48 


7. 65 


18. 41 


7. 81 


18. 34 


7.97 


18. 27 


8. 13 




21 


19. 40 


8. 04 


19. 33 


8. 21 


19. 26 


8.37 


19. 18 


8. 54 




22 


20. 33 


8. 42 


20. 25 


8. 60 


20. 18 


8.77 


20. 10 


8. 95 




23 


21. 25 


8. 80 


21. 17 


8. 99 


21. 09 


9. 17 


21. 01 


9. 35 




24 


22. 17 


9. 18 


22. 09 


9. 38 


22. 01 


9.57 


21. 93 


9. 76 




25 


23. 10 


9. 57 


23. 01 


9. 77 


22. 93 


9.97 


22. 84 


10. 17 




26 


24. 02 


9. 95 


23. 93 


10. 16 


23. 84 


10. 37 


23. 75 


10. 58 




27 


24. 94 


10. 33 


24. 85 


10. 55 


24. 76 


10.77 


24. 67 


10. 98 




28 


25. 87 


10. 72 


25. 77 


10. 94 


25. 68 


11. 16 


25. 58 


11. 39 




29 


26. 79 


11. 10 


26. 69 


11. 33 


26. 59 


11.56 


26. 49 


11. 80 




30 


27. 72 


11. 48 


27. 62 


11. 52 


27.51 


11.96 


27. 41 


12. 20 




35 


32. 34 


13. 39 


32. 22 


13- 68 


32. 10 


13.96 


31. 97 


14. 24 




40 


36. 96 


15. 31 


36. 82 


15. 63 


36. 68 


15.95 


36. 54 


16. 27 




45 


41. 57 


17. 22 


41. 42 


17. 58 


41. 27 


17.94 


41. 11 


18. 30 




50 


46. 19 


19. 13 


46. 03 


19. 54 


45. 85 


19.94 


45. 68 


20. 34 




55 


50. 81 


21. 05 


50. 63 


21. 49 


50. 44 


21.93 


50. 24 


22. 37 




60 


55. 43 


22. 96 


55. 23 


23. 44 


55. 02 


23.92 


54. 81 


24. 40 




65 


60. 05 


24. 87 


59. 83 


25. 40 


59. 61 


25.92 


59. 38 


26. 44 




70 


64. 67 


26. 79 


64. 44 


27. 35 


64. 19 


27.91 


63. 95 


28. 47 




75 


69. 29 


28. 70 


69. 04 


29. 30 


68. 78 


29.91 


68. 52 


30. 51 




80 


73. 91 


30. 61 


73. 64 


31. 26 


73. 36 


31.90 


73. 08 


32. 54 




85 


78. 53 


32. 53 


78. 24 


33. 21 


77. 95 


33.89 


77. 65 


34. 57 




90 


83. 15 


34. 44 


82. 85 


35. 17 


82. 54 


35.89 


82. 22 


36. 61 




95 


87. 77 


36. 35 


87. 45 


37. 12 


87. 12 


37.88 


86. 79 


38. 64 




100 


92. 39 


38. 27 


92. 05 


39. 07 


91.71 


39.87 


91. 35 


40. 67 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




67% Beg. 


67 Deg. 


66% Deg. 


66 ] 


Deg. 











TRAVERSE TABLE. 


83 




s 

1 


24^ Deg. 


25 Deg. 


25^ Deg- ' 


26 1 


eg. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




1 


0. 91 


0. 41 


0. 91 


0. 42 


0. 90 


0. 43 


0. 90 


0. 44 




2 


1. 82 


0. 83 


1. 81 


0. 85 


1. 81 


0. 86 


1. 80 


0. 88 




3 


2. 73 


1. 24 


2. 72 


1. 27 


2. 71 


1. 29 


2. 70 


1. 32 




4 


3. 64 


1. 66 


3. 63 


1. 69 


3. 61 


1. 72 


3. 60 


1. 75 




5 


4. 55 


2. 07 


4.53 


2. 11 


4. 51 


2. 15 


4. 49 


2. 19 




6 


5. 46 


2. 49 


5. 44 


2. 54 


5. 42 


2. 58 


5. 39 


2. 63 




7 


6. 37 


2. 90 


6. 34 


2. 96 


6. 32 


3. 01 


6. 29 


3. 07 




8 


7. 28 


3. 32 


7. 25 


3. 38 


7. 22 


3. 44 


7. 19 


3. 51 




9 


8. -19 


3. 73 


8. 16 


3. 80 


8. 12 


3. 87 


8. 09 


3. 95 




10 


9. 10 


4. 15 


9. 06 


4. 23 


9. 03 


4. 31 


8. 99 


4. 38 




11 


10. 01 


4. 56 


9. 97 


4. 65 


9. 93 


4. 74 


9. 89 


4. 82 




12 


10. 92 


4. 98 


10. 88 


5. 07 


10. 83 


5. 17 


10. 79 


5. 26 




13 


11. 83 


5. 39 


11. 78 


5. 49 


11. 73 


5. 60 


11. 68 


5. 70 




14 


12. 74 


5. 81 


12. 69 


5. 92 


12. 64 


6. 03 


12. 58 


6. 14 




15 


13. 65 


6. 22 


13. 59 


6. 34 


13. 54 


6. 46 


13. 48 


6. 58 




16 


14. 56 


6. 64 


14. 50 


6. 76 


14. 44 


6. 89 


14. 38 


7. 01 




17 


15. 47 


7. 05 


15. 41 


7. 18 


15. 34 


7. 32 


15. 28 


7. 45 




18 


16. 38 


7. 46 


16. 31 


7. 61 


16. 25 


7. 75 


16. 18 


7. 89 




19 


17. 19 


7. 88 


17. 22 


8. 03 


17. 15 


8- 18 


17. 08 


8. 33 




20 


18. 20 


8. 29 


18. 13 


8. 45 


18. 05 


8- 61 


17. 98 


8. 77 




21 


19. 11 


8. 71 


19. 03 


8. 87 


18. 95 


9- 04 


18. 87 


9. 21 




22 


20. 02 


9. 12 


19. 94 


9. 30 


19. 86 


9- 47 


19. 77 


9. 64 




23 


20. 93 


9. 54 


20. 85 


9. 72 


20. 76 


9. 90 


20. 67 


10. 08 




24 


21. 84 


9. 95 


21. 75 


10. 14 


21. 66 


10- 33 


21. 57 


10. 52 




25 


22. 75 


10. 37 


22. 66 


10. 57 


22. 56 


10- 76 


22. 47 


10. 96 




26 


23. 66 


10. 78 


23. 56 


10. 99 


23. 47 


11- 19 


23. 37 


11. 40 




27 


24. 57 


11. 20 


24. 47 


n. 41 


24- 37 


11- 62 


24. 27 


11. 84 




28 


25. 48 


11. 61 


25. 38 


H. 83 


25- 27 


12- 05 


25. 17 


12. 27 




29 


26. 39 


12. 03 


26. 28 


12. 26 


26- 17 


12- 48 


26. 06 


12. 71 




30 


27. 30 


12. 44 


27. 19 


12. 68 


27. 08 


12- 92 


26. 96 


13. 15 




35 


31. 85 


14. 51 


31. 72 


14. 79 


31- 59 


15- 07 


31. 46 


15. 34 




40 


36. 40 


16. 59 


36. 25 


16. 90 


36. 10 


17- 22 


35. 95 


17. 53 




45 


40. 95 


18. 66 


40. 78 


19. 02 


40- 62 


19. 37 


40. 45 


19. 73 




50 


45. 50 


20. 73 


45. 32 


21. 13 


45. 13 


21- 53 


44. 94 


21. 92 




55 


50. 05 


22. 81 


49. 85 


23. 24 


49. 64 


23- 68 


49. 43 


24. 11 




60 


54. 60 


24. 88 


54. 38 


25. 36 


54. 16 


25- 83 


53. 93 


26. 30 




65 


59. 15 


26. 96 


58. 91 


27 47 


58. 67 


27. 98 


58. 42 


28. 49 




70 


63. 70 


29. 03 


63. 44 


29. 58 


63. 18 


30. 14 


62. 92 


30. 69 




75 


68. 25 


31. 10 


67. 97 


31. 70 


67. 69 


32. 29 


67. 41 


32. 88 




80 


72. 80 


33. 18 


72. 50 


33. 81 


72. 21 


34. 44 


71. 90 


35. 07 




85 


77. 35 


35. 25 


77. 04 


35. 92 


76. 72 


36. 59 


76. 40 


37. 26 




90 


81. 90 


37. 32 


81. 57 


38. 04 


81. 23 


38. 75 


80. 89 


39. 45 




95 


86. 45 


39. 40 


86. 10 


40. 15 


85. 75 


40. 90 


85. 39 


41. 65 




100 


91. 00 


41.47 

Lat. 


90. 63 


42. 26 


90. 26 


43. 05 


89. 88 


43. 84 




Dep. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




65^ Deg. 


65 Deg. 


64^ Deg. 


64 Deg. 




I— — 





84 


TRAVERSE TABLE. 




g 

S' 

I 
8 

1 


26^ Deg. 


27 Deg. 


27K Deg. 


28 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 
0.46 


Lat. 


Dep. 


0. 89 


0.45 


0. 89 


0. 45 


0. 89 


0. 88 


0. 47 


2 


1. 79 


0. 89 


1.78 


0. 91 


1. 77 


0.92 


1.77 


0. 94 


3 


2. 68 


1. 34 


2. 67 


1. 36 


2. 66 


1.39 


2. 65 


1. 41 


4 


3. 58 


1. 78 


3. 56 


1. 82 


3. 55 


1.85 


3. 53 


1. 88 


5 


4. 57 


2. 23 


4. 45 


2. 27 


4. 44 


2.31 


4. 41 


2. 35 


6 


5. 37 


2. 68 


5. 35 


2. 72 


5- 32 


2.77 


5. 30 


2. 82 


7 


6.26 


3. 12 


6. 24 


3. 18 


6- 21 


3.23 


6. 18 


3.29 


8 


7. 16 


3. 47 


7. 13 


3. 63 


7- 10 


3.69 


7. 06 


3. 76 


9 


8. 05 


4. 02 


8. 02 


4. 09 


7- 98 


4. 16 


7. 95 


4. 23 


10 


8. 95 


4. 46 


8. 91 


4. 54 


8- 87 


4.62 


8. 83 


4. 69 


11 


9. 84 


4. 91 


9. 80 


4. 99 


9. 76 


5.08 


9 7l 


5. 16 


12 


10. 74 


5. 35 


10. 69 


5. 45 


10- 64 


5.54 


10. 60 


5. 63 


13 


11. 63 


5. 80 


11. 58 


5. 90 


11-53 


6.00 


11. 48 


6. 10 


14 


12. 53 


6. 25 


12. 47 


6. 36 


12- 42 


6.49 


12. 36 


6.57 


15 


13. 42 


6. 69 


13. 37 


6. 81 


13- 31 


6.93 


13. 24 


7. 04 


16 


14. 32 


7. 14 


14. 26 


7.26 


14- 19 


7.39 


14. 13 


7. 51 


17 


15. 21 


7. 59 


15. 15 


7.72 


15- 08 


7.85 


15. 01 


7. 98 


18 


16. 11 


8. 03 


16. 04 


8. 17 


15. 97 


8.31 


15. 89 


8. 45 


19 


17. 00 


8. 48 


16. 93 


8. 63 


16- 85 


8.77 


16. 78 


8. 92 


20 


17. 90 


8. 92 


17. 82 


9. 08 


17- 74 


9.23 


17. 66 


9. 39 


21 


18. 79 


9. 37 


18. 71 


9. 53 


18- 63 


9.70 


18. 54 


9. 86 


22 


19. 69 


9. 82 


19. 60 


9. 99 


19- 51 


10. 16 


19. 42 


10. 33 


23 


20. 58 


10. 26 


20. 49 


10. 44 


20- 40 


10.62 


20. 31 


10. 80 


24 


21. 48 


10. 71 


21. 38 


10. 90 


21- 29 


11.08 


21. 19 


11. 27 


25 


22. 37 


11. 15 


22. 28 


11. 35 


22- 18 


11.54 


22. 07 


11. 74 


26 


23. 27 


11. 60 


23. 17 


11. 80 


23- 06 


12.01 


22. 96 


12. 21 


27 


24. 16 


12. 05 


24. 06 


12. 26 


23- 95 


12.47 


23. 84 


12. 68 


28 


25. 06 


12. 49 


24. 95 


12- 71 


24- 84 


12.93 


24. 72 


13. 15 


29 


25. 95 


12. 94 


25. 84 


13. 17 


25- 72 


13.39 


25. 61 


13. 61 


30 


26. 85 


13. 39 


26. 73 


13- 62 


26- 61 


13.85 


26. 49 


14. 08 


35 


31. 32 


15. 62 


31. 19 


15- 89 


31- 05 


16.16 


30. 90 


16. 43 


40 


35. 80 


17. 85 


35. 64 


18. 16 


35- 48 


18.47 


35. 32 


18. 78 


45 


40. 27 


20. 08 


40. 10 


20. 43 


39- 92 


20.78 


39. 73 


21. 13 


50 


44. 75 


22. 31 


44. 55 


22. 70 


44- 35 


23.09 


44. 15 


23. 47 


55 


49. 22 


24. 54 


49. 01 


24. 97 


48. 79 


25.40 


48. 56 


25. 82 


60 


53. 70 


26. 77 


53. 46 


27. 24 


53. 22 


27.70 


52. 98 


28. 17 


65 


58. 17 


29. 00 


57. 92 


29. 51 


57. 66 


30.01 


57. 39 


30. 52 


70 


62. 65 


31. 23 


62. 37 


31. 78 


62. 09 


32.32 


61. 81 


32. 86 


75 


67. 12 


33. 46 


66. 83 


34. 05 


66. 53 


34.63 


66. 22 


35. 21 


80 


71. 59 


35. 70 


71. 28 


36. 32 


70. 96 


36.94 


70. 64 


37. 56 


85 


76. 07 


37. 93 


75. 74 


38. 59 


75. 40 


39.25 


75. 05 


39. 91 


90 


80. 54 


40. 16 


80. 19 


40. 86 


79. 83 


41.56 


79. 47 


42. 25 


95 


85. 02 


42. 39 


84. 65 


43. 13 


84. 27 


43.87 


83. 88 


44. 60 


100 


89. 49 


44. 62 


89. 10 


45. 40 


88. 90 


46. 17 


88. 29 


46.95 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


63^ Deg. 


63 


Deg. 


62^ Deg. 


62 


Deg. 















TRAVERSE TABLE. 


1 

85 




s 

53° 
1 
P 


28K ©eg. 


29 Deg. 


29^ Deg. 


30 Deg. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat 


Dep. 




1 


0. 88 


0. 48 


0. 87 


0. 48 


0. 87 


0. 49 


0. 87 


0. 50 




2 


1. 76 


0. 95 


1.75 


0. 97 


1. 74 


0. 98 


1. 73 


1. 00 




3 


2. 64 


1. 43 


2. 62 


1.45 


2. 61 


1. 48 


2. 60 


1. 50 




4 


3. 52 


1.91 


3. 50 


1. 94 


3. 48 


1. 97 


3. 46 


2. 00 




5 


4. 39 


2. 39 


4. 37 


2. 42 


4. 35 


2. 46 


4. 33 


2. 50 




6 


5. 27 


2. 86 


5. 25 


2. 91 


5. 22 


2. 95 


5. 20 


3. 00 




7 


6. 15 


3. 34 


6. 12 


3. 39 


6. 09 


3. 45 


6. 06 


3. 50 




8 


7. 03 


3. 82 


7. 00 


3. 88 


6. 96 


3. 94 


6. 93 


4. 00 




9 


7. 91 


4. 29 


7. 87 


4. 36 


7. 83 


4. 43 


7. 79 


4. 50 




10 


8. 79 


4. 77 


8. 75 


4. 85 


8. 70 


4. 92 


8. 66 


5. 00 




11 


9. 67 


5. 25 


9. 62 


5. 33 


9. 57 


5. 42 


9. 53 


5. 50 




12 


10. 55 


5. 73 


10. 50 


5. 82 


10. 44 


5. 91 


10. 39 


6. 00 




13 


11. 42 


6. 20 


11. 37 


6. 30 


11. 31 


6. 40 


11. 26 


6. 50 




14 


12. 30 


6. 68 


12. 24 


6. 79 


12. 18 


6. 89 


12. 12 


7. 00 




15 


13. 18 


7. 16 


13. 12 


7. 27 


13. 06 


7. 39 


12. 99 


7. 50 




16 


14. 06 


7. 63 


13. 99 


7. 76 


13. 93 


7. 88 


13. 86 


8. 00 




17 


14. 94 


8. 11 


14. 87 


8. 24 


14. 80 


8- 37 


14. 72 


8. 50 




18 


15. 82 


8. 59 


15. 74 


8. 73 


15. 67 


8. 86 


15. 59 


9. 00 




19 


16. 70 


9. 07 


16. 62 


9. 21 


16. 54 


9. 36 


16. 45 


9. 50 




20 


17. 58 


9. 54 


17. 49 


9. 70 


17. 41 


9. 85 


17. 32 


10. 00 




21 


18. 46 


10. 02 


18. 37 


10. 18 


18. 28 


10- 34 


18. 19 


10. 50 




22 


19. 33 


10. 50 


19. 24 


10. 67 


19. 15 


10. 83 


19. 05 


11. 00 




23 


20. 21 


10. 97 


20. 12 


11. 15 


20. 02 


11. 33 


19. 92 


11. 50 




24 


21. 09 


11. 45 


20. 99 


11. 64 


20. 89 


11. 82 


20. 78 


12. 00 




25 


21. 97 


11. 93 


21. 87 


12. 12 


21. 76 


12. 31 


21. 65 


12. 50 




26 


22. 85 


12. 41 


22. 74 


12. 60 


22. 63 


12- 80 


22. 52 


13. 00 




27 


23. 73 


12. 88 


23. 61 


13. 09 


23. 50 


13. 30 


23. 38 


13. 50 




28 


24. 61 


13. 36 


24. 49 


13. 57 


24. 37 


13. 79 


24. 25 


14. 00 




29 


25. 49 


13. 84 


25. 36 


14. 06 


25. 24 


14. 28 


25. 11 


14. 50 




30 


26. 36 


14. 31 


26. 24 


14. 54 


26. 11 


14-77 


25. 98 


15. 00 




35 


30. 76 


16. 70 


30. 61 


16. 97 


30. 46 


17- 23 


30. 31 


17. 50 




40 


35. 15 


19. 09 


34. 98 


19. 39 


34. 81 


19. 70 


34. 64 


20. 00 




45 


39. 55 


21. 47 


39. 36 


21. 82 


39. 17 


22. 16 


38. 97 


22. 50 




50 


43. 94 


23. 86 


43. 73 


24. 24 


43. 52 


24. 62 


43. 30 


25. 00 




55 


48. 33 


26. 24 


48. 10 


26. 66 


47. 87 


27. 08 


47. 63 


27. 50 




60 


52. 73 


28. 63 


52. 48 


29. 09 


52. 22 


29. 55 


51. 96 


30. 00 




65 


57. 12 


31. 02 


56. 85 


31. 51 


56. 57 


32. 01 


56. 29 


32. 50 




70 


61. 52 


33. 40 


61. 22 


33. 94 


60. 92 


34. 47 


60. 62 


35. 00 




75 


65. 91 


35. 79 


65. 60 


36. 36 


65. 28 


36. 93 


64. 95 


37. 50 




80 


70. 31 


38. 17 


69. 97 


38. 78 


69. 63 


39. 39 


69. 28 


40. 00 




85 


74. 70 


40. 56 


74. 34 


41. 21 


73. 98 


41. 86 


73. 61 


42. 50 




90 


79. 09 


42. 94 


78. 72 


43. 63 


78. 33 


44. 32 


77. 94 


45. 00 




95 


83. 49 


45. 33 


83. 09 


46. 06 


82. 68 


46. 78 


82. 27 


47. 50 




100 


87. 88 


47. 72 


87. 46 


48. 48 


87. 04 


49. 24 


86. 60 


50. 00 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




61^ 


Deg. 


61] 


Deg. 


60^ 


Deg. 


60 


Deg. 




1 , 





86 




TRAVERSE TABLE. 






55 
1 

8 

1 


30^ Deg. 


31 Deg. 


31^ Deg. 


32 Deg. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 
0. 85 


Dep. 
0. 53 




0. 86 


0.51 


0. 86 


0. 51 


0. 85 


0.52 




2 


1. 72 


1. 02 


1.71 


1. 03 


1. 71 


1. 04 


1. 70 


1. 06 




3 


2. 58 


1. 52 


2. 57 


1. 55 


2.56 


1.57 


2. 54 


1. 59 




4 


3. 45 


2. 03 


.3. 43 


2. 06 


3. 41 


2. 09 


3. 39 


2. 12 




5 


4. 31 


2. 54 


4.29 


2. 58 


4. 26 


2. 61 


4. 24 


2. 65 




6 


5. 17 


3. 05 


5. 14 


3. 09 


5- 12 


3. 13 


5. 09 


3. 18 




7 


6. 03 


3. 55 


6. 00 


3. 61 


5. 97 


3. 66 


5. 94 


3. 71 




8 


6. 89 


4. 06 


6. 86 


4. 12 


6. 82 


4. 18 


6. 78 


4. 24 




9 


7. 75 


4. 57 


7. 71 


4. 64 


7. 67 


4. 70 


7. 63 


4. 77 




10 


8. 62 


5. 08 


8. 57 


5. 15 


8.53 


5. 22 


8. 48 


5. 30 




11 


9. 48 


5. 58 


9. 43 


5. 67 


9. 38 


5. 75 


9 33 


5. 83 




12 


10. 34 


6. 09 


10. 29 


6. 18 


10- 23 


6. 27 


10. 18 


6. 36 




13 


11. 20 


6.60 


11. 14 


6. 70, 


11. 08 


6. 79 


11. 02 


6. 89 




14 


12. 06 


7. 11 


12. 00 


7.21 


11. 94 


7. 31 


11. 87 


7. 42 




15 


12. 92 


7. 61 


12. 86 


7. 73 


12- 79 


7. 84 


12. 72 


7. 95 




16 


13. 79 


8. 12 


13. 71 


8- 24 


13- 64 


8. 36 


13. 57 


8. 48 




17 


14. 65 


8. 63 


14. 57 


8- 77 


14- 49 


8. 88 


14. 42 


9. 01 




18 


15. 51 


9. 14 


15. 43 


9- 27 


15. 35 


9. 40 


15. 26 


9. 54 




19 


16. 37 


9. 64 


16. 29 


9- 79 


16. 20 


9. 93 


16. 11 


10. 07 




20 


17. 23 


10. 15 


17. 14 


10- 30 


17. 05 


10. 45 


16. 96 


10. 60 




21 


18. 09 


10. 66 


18. 00 


10- 82 


17. 91 


10. 97 


17. 81 


11. 13 




22 


18. 96 


11. 17 


18. 86 


11. 33 


18. 76 


11. 49 


18. 66 


11.66 




23 


19. 82 


11. 67 


19. 71 


11. 85 


19. 61 


12. 02 


19. 51 


12. 19 




24 


20. 68 


12. 18 


20. 57 


12. 36 


20- 46 


12. 54 


20. 35 


12. 72 




25 


21. 54 


12. 69 


21. 43 


12- 88 


21. 32 


13. 06 


21. 20 


13.25 




26 


22. 40 


13. 20 


22. 29 


13. 39 


22. 17 


13. 58 


22. 05 


13. 78 




27 


23. 26 


13. 70 


23. 14 


13. 91 


23- 02 


14. 11 


22. 90 


14. 31 




28 


24. 13 


14. 21 


24. 00 


14- 42 


23. 87 


14. 63 


23. 75 


14. 84 




29 


24. 99 


14. 72 


42. 86 


14- 94 


24. 73 


15. 15 


24. 59 


15. 37 




30 


25.85 


15. 23 


25. 71 


15- 45 


25. 58 


15. 67 


25. 44 


15.90 




35 


30. 16 


17. 76 


30. 00 


18. 03 


29. 84 


18. 29 


29. 68 


18. 55 




40 


34. 47 


20. 30 


34. 29 


20- 60 


34. 11 


20. 90 


33. 92 


21.20 




45 


38. 77 


22. 84 


38. 57 


23- 18 


38. 37 


23. 51 


38. 16 


23. 85 




50 


43. 08 


25. 38 


42. 86 


25. 75 


42. 63 


26. 12 


42. 40 


26. 50 




55 


47. 39 


27. 91 


47. 14 


28. 33 


46. 90 


28. 74 


46. 64 


29. 15 




60 


51. 70 


30. 45 


51. 53 


30. 90 


51. 16 


31. 35 


50. 88 


31. 80 




65 


56. 01 


32. 99 


55. 72 


33- 48 


55. 42 


33. 96 


55. 12 


34. 44 




70 


60. 31 


35. 53 


60. 00 


36. 05 


59. 68 


36.57 


59. 36 


37. 09 




75 


64. 62 


38. 07 


64. 29 


38. 63 


63. 95 


39. 19 


63. 60 


39. 74 




80 


68. 93 


40. 60 


68. 57 


41. 20 


68. 21 


41. 80 


67. 84 


42. 39 




85 


73. 24 


43. 14 


72. 86 


43. 78 


72. 47 


44. 41 


72. 08 


45. 04 




90 


77. 55 


45. 68 


77. 15 


46. 35 


76. 74 


47. 02 


76. 32 


47. 69 




95 


81. 85 


48. 22 


81. 43 


48. 93 


81. 00 


49. 64 


80. 56 


50. 34 




100 


86. 16 


50.75 


85. 72 


51. 50 


85. 26 


52. 25 


84. 80 


52. 59 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




59^ Deg. 


59 


Deg. 


58^ Leg. 


58 Deg. 











TRAVERSE TABLE. 




87 




1 
5 


32^ Deg. 


33 Deg. 


33^ Deg. 




34] 


Deg. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




1 


0. 84 


0. 54 


0. 84 


0. 54 


0. 83 


0. 55 


0. 


83 


0.56 




2 


1. 69 


1. 07 


1. 68 


1. 09 


1. 67 


1. 10 


1. 


66 


1. 12 




3 


2. 53 


1. 61 


2. 52 


1. 63 


2. 50 


1. 66 


2. 


49 


1. 68 




4 


3. 37 


2. 15 


3. 35 


2. 18 


3. 34 


2. 21 


3. 


32 


2. 24 




5 


4. 22 


2. 69 


4. 19 


2. 72 


4. 17 


2. 76 


4. 


15 


2. 80 




6 


5. 06 


3. 22 


5. 03 


3. 27 


5. 00 


3. 31 


4. 


97 


3. 36 




7 


5. 90 


3. 76 


5. 87 


3. 81 


5. 84 


3. 86 


5. 


80 


3. 91 




8 


6. 75 


4. 30 


6. 71 


4. 36 


6. 67 


4. 42 


6. 


63 


4. 47 




9 


7. 59 


4. 84 


7. 55 


4. 90 


7. 50 


4. 97 


7. 


46 


5. 03 




10 


8. 43 


5. 37 


8. 39 


5. 45 


8. 34 


5. 52 


8. 


29 


5. 59 




11 


9. 28 


5. 91 


9. 23 


5. 99 


9. 17 


6. 07 


9. 


12 


6. 15 




12 


10. 12 


6. 45 


10. 06 


6. 54 


10. 01 


6. 62 


9. 


95 


6. 71 




13 


10. 96 


6. 98 


10. 90 


7. 08 


10. 84 


7. 18 


10. 


78 


7. 27 




14 


11. 81 


7. 52 


11. 74 


7. 62 


11. 67 


7.73 


11. 


61 


7. 83 




15 


12. 65 


8. 06 


12. 58 


8. 17 


12. 51 


8. 28 


12. 


44 


8. 39 




16 


13. 49 


8. 60 


13. 42 


8. 71 


13. 34 


8. 83 


13. 


26 


8. 95 




17 


14. 34 


9. 13 


14. 26 


9. 26 


14. 18 


9. 38 


14. 


09 


9. 51 




18 


15. 18 


9. 67 


15. 10 


9. 80 


15. 01 


9. 93 


14. 


92 


10. 07 




19 


16. 02 


10. 21 


15. 93 


10. 35 


15. 84 


10- 49 


15. 


75 


10. 62 




20 


16. 87 


10. 75 


16. 77 


10. 89 


16. 68 


11. 04 


16. 


58 


11. 18 




21 


17. 71 


11.28 


17. 61 


11. 44 


17. 51 


11. 59 


17. 


41 


11. 74 




22 


18. 55 


11. 82 


18. 45 


11. 98 


18. 35 


12- 14 


18.. 24 


12. 30 




23 


19. 40 


12. 36 


19. 29 


12. 53 


19. 18 


12. 69 


19. 


07 


12. 86 




24 


20. 28 


12. 90 


20. 13 


13. 07 


20. 01 


13- 25 


19. 


90 


13. 42 




25 


21. 08 


13. 43 


20. 97 


13. 62 


20. 85 


13- 80 


20. 


73 


13. 98 




26 


21. 93 


13. 97 


21. 81 


14. 16 


21. 68 


14- 35 


21. 


55 


14. 54 




27 


22. 77 


14. 51 


22. 64 


14.71 


22. 51 


14. 90 


22. 


38 


15. 10 




28 


23. 61 


15. 04 


23. 48 


15. 25 


23. 35 


15. 45 


23. 


21 


15. 66 




29 


24. 46 


15.58 


24. 32 


15. 97 


24. 18 


16- 01 


24. 


04 


16. 22 




30 


25. 30 


16. 12 


25. 16 


16. 34 


25. 02 


16. 56 


24. 


87 


16. 78 




35 


29. 52 


18. 81 


29. 35 


19. 06 


29. 19 


19. 32 


29. 


02 


19. 57 




40 


33. 74 


21. 49 


33. 55 


21. 79 


33. 36 


22. 08 


33. 


16 


22. 37 




45 


37. 95 


24. 18 


37. 74 


24. 51 


37. 52 


24. 84 


37. 


31 


25. 16 




50 


42. 17 


26. 86 


41. 93 


27. 23 


41. 69 


27. 60 


41. 


45 


27. 96 




55 


46. 39 


29. 55 


46. 13 


29. 96 


45. 86 


30. 36 


45. 


60 


30. 76 




60 


50. 60 


32. 24 


50. 32 


32. 68 


50. 08 


33. 12 


49. 


74 


33. 55 




65 


54. 82 


34. 92 


54. 51 


35. 40 


54. 20 


35. 88 


53. 


89 


36. 35 




70 


59. 04 


37. 61 


58. 72 


38. 12 


58. 37 


38. 64 


58. 


03 


39. 14 




75 


63. 25 


40. 30 


62. 90 


40. 85 


62. 54 


41. 40 


62. 


18 


41. 94 




80 


67. 47 


42. 98 


67. 09 


43. 57 


66. 71 


44. 15 


66. 


32 


44. 74 




85 


71. 69 


45. 67 


71 29 


46. 29 


70. 88 


46. 91 


70. 


47 


47. 53 




90 


75. 91 


48. 36 


75. 48 


49. 02 


75. 05 


49. 67 


74. 


61 


50. 33 




95 


80. 12 


51. 04 


79. 67 


51. 74 


79. 22 


54. 43 


78. 


76 


53. 12 




100 


84. 34 


53. 73 


83. 87 


54. 46 


83. 39 


55. 19 


82. 


90 


55. 92 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




W% 


Deg. 


57 Deg. 


56^ Deg. 




56] 


Deg. 












! 


1 



88 






TRAVERSE TABLE. 




p 
p 

o 
CD 


34^ »eg. 


35 ] 


Deg. 


35^ Deg. 


36 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 

0. 57 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0. 


82 


0. 


57 


0. 82 


0. 81 


0. 58 


0. 81 


0. 59 


2 


1. 


65 


1. 


13 


1. 64 


1. 15 


1. 63 


1. 16 


1. 62 


1. 18 


3 


2. 


47 


1. 


70 


2. 46 


1. 72 


2. 44 


1. 74 


2. 43 


1. 76 


4 


3. 


30 


2. 


27 


3. 28 


2. 29 


3. 26 


2. 32 


3. 24 


2. 35 


5 


4. 


12 


2. 


83 


4. 10 


2. 87 


4. 07 


2. 90 


4. 05 


2. 94 


6 


4. 


94 


3. 


40 


4. 91 


3. 44 


4. 88 


3. 48 


4. 85 


3. 53 


7 


5. 


77 


3. 


96 


5. 73 


4. 01 


5. 70 


4. 06 


5. 66 


4. 11 


8 


6. 


59 


4. 


53 


6. 55 


4. 59 


6. 51 


4. 65 


6. 47 


4. 70 


9 


7. 


42 


5. 


10 


7. 37 


5. 16 


7. 33 


5. 23 


7. 28 


5. 29 


10 


8. 


24 


5. 


66 


8. 19 


5. 74 


8. 14 


5. 81 


8. 09 


5. 88 


11 


9. 


07 


6. 


23 


9. 01 


6. 31 


8. 96 


6. 39 


8. 90 


6. 47 


12 


9. 


89 


6. 


80 


9. 83 


6. 88 


9. 77 


6. 97 


9. 71 


7. 05 


13 


10. 


71 


7. 


36 


10. 65 


7. 46 


10- 58 


7. 55 


10. 52 


7. 64 


14 


11. 


54 


7. 


93 


11. 47 


8. 03 


11. 40 


8. 13 


11. 33 


8. 23 


15 


12. 


36 


8. 


50 


12. 29 


8. 60 


12. 21 


8. 71 


12. 14 


8. 82 


16 


13. 


19 


9. 


06 


13. 11 


9. 18 


13. 03 


9. 29 


12. 94 


9. 40 


17 


14. 


01 


9. 


63 


13. 93 


9. 75 


13. 84 


9. 87 


13. 75 


9. 99 


18 


14. 


83 


10. 


20 


14. 74 


10. 32 


14. 65 


10. 45 


14. 56 


10. 58 


19 


15. 


66 


10. 


76 


15. 56 


10. 90 


15. 47 


11. 03 


15. 37 


11. 17 


20 


16. 


48 


11. 


33 


16. 38 


11. 47 


16. 28 


11. 61 


16. 18 


11. 76 


21 


17. 


31 


11. 


89 


17. 20 


12. 05 


17. 10 


12. 19 


16. 99 


12. 34 


22 


18. 


13« 


12. 


46 


18. 02 


12. 62 


17. 91 


12. 78 


17. 80 


12. 93 


23 


18. 


95 


13. 


03 


18. 84 


13. 19 


18. 72 


13. 36 


18. 61 


13. 52 


24 


19. 


78 


13. 


59 


19. 66 


13. 77 


19. 54 


13. 94 


19. 42 


14. 11 


25 


20. 


60 


14. 


16 


20. 48 


14. 34 


20. 35 


14. 52 


20. 23 


14. 69 


26 


21. 


43 


14. 


73 


21. 30 


14. 91 


21. 17 


15. 10 


21. 03 


15. 28 


27 


22. 


25 


15. 


29 


22. 12 


15. 49 


21. 98 


15. 68 


21. 84 


15. 87 


28 


23. 


08 


15. 


86 


22. 94 


16. 06 


22. 80 


16. 26 


22. 65 


16. 46 


29 


23. 


90 


16. 


43 


23. 76 


16. -63 


23. 61 


16. 84 


23. 46 


17. 05 


30 


24. 


72 


16. 


99 


24. 57 


17.21 


24. 42 


17. 42 


24. 27 


17. 63 


35 


28. 


84 


19. 


82 


28. 67 


20. 08 


28. 49 


20. 32 


28. 32 


20.57 


40 


32. 


97 


22. 


66 


32. 77 


22. 94 


32. 56 


23. 23 


32. 36 


23. 51 


45 


37. 


09 


25. 


49 


36. 86 


25. 81 


36. 64 


26. 13 


36. 41 


26. 45 


50 


41. 


21 


28. 


32 


40. 96 


28. 68 


40. 71 


29. 04 


40. 45 


29. 39 


55 


45. 


33 


31. 


15 


45. 05 


31.55 


44. 78 


31. 94 


44. 50 


32. 23 


60 


49. 


45 


33. 


98 


49. 15 


34. 41 


48. 85 


34. 84 


48. 54 


35. 27 


65 


53. 


57 


36. 


82 


53. 24 


37. 28 


52. 92 


37. 75 


52. 59 


38. 21 


70 


57. 


69 


39. 


65 


57. 34 


40. 15 


56. 99 


40. 65 


56. 63 


41. 14 


75 


61. 


81 


42. 


48 


61. 44 


43. 02 


61. 06 


43. 55 


60. 68 


44. 08 


80 


65. 


93 


45. 


31 


65. 53 


45. 89 


65. 13 


46. 46 


64. 72 


47. 02 


85 


70. 


05 


48. 


14 


69. 63 


48. 75 


69. 20 


49. 36 


68. 77 


49. 96 


90 


74. 


17 


50. 


98 


73. 72 


51. 62 


73. 27 


52. 26 


72. 81 


52. 90 


95 


78. 


29 


53. 


81 


77. 82 


54. 49 


77. 34 


55. 17 


76. 86 


55. 84 


100 


82. 


41 


56. 


64 


81. 92 


57. 36 


81. 41 


58. 07 


80. 90 


58. 78 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


55^ Deg. 


55 ] 


Deg. 


54^ Deg. 


54 Deg. 











TRAVERSE TABLE. 


89 




1 


36^ Deg. 


37 Deg. 


37K Deg. 


38 Deg. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




1 


0. 80 


0. 


59 


0. 


80 


0. 


60 


0. 79 


0. 61 


0. 79 


0. 62 




2 


1. 61 


1. 


19 


1. 


60 


1. 


20 


1. 59 


1. 22 


1. 58 


1. 23 




3 


2. 41 


1. 


78 


2. 


40 


1. 


81 


2. 38 


1. 83 


2. 36 


1. 85 


! 


4 


3. 22 


2. 


38 


3. 


19 


2. 


41 


3. 17 


2- 43 


3. 15 


2. 46 




5 


4. 02 


2. 


97 


3. 


99 


3. 


01 


3. 97 


3- 04 


3. 94 


3. 08 




6 


4. 82 


3. 


57 


4. 


79 


3. 


61 


4. 76 


3- 65 


4. 73 


3. 69 




7 


5. 63 


4. 


16 


5. 


59 


4. 


21 


5. 55 


4- 20 


5. 52 


4. 31 




8 


6. 43 


4. 


76 


6. 


39 


4. 


81 


6. 35 


4- 87 


6. 30 


4. 93 




9 


7. 23 


5. 


35 


7. 


19 


5. 


42 


7. 14 


5- 48 


7. 09 


5. 54 




10 


8. 04 


5. 


95 


7. 


99 


6- 


02 


7. 93 


6- 09 


7. 88 


6. 16 




11 


8. 84 


6. 


54 


8. 


78 


6. 


62 


8. 73 


6. 70 


8. 67 


6. 77 




12 


9. 65 


7. 


14 


9. 


58 


7. 


22 


9.52 


7. 31 


9. 46 


7. 39 




13 


10. 45 


7. 


73 


10. 


38 


7. 


82 


10. 31 


7- 91 


10. 24 


8. 00 




14 


11. 25 


8. 


33 


11. 


18 


8. 


43 


11. 11 


8- 52 


11. 03 


8. 62 




15 


12. 06 


8. 


92 


11. 


98 


9. 


03 


11. 90 


9- 13 


11. 82 


9. 23 




16 


12. 86 


9. 


52 


12. 


78 


9. 


63 


12. 69 


9-74 


12. 61 


9. 85 




17 


13. 67 


10. 


11 


13. 


58 


10. 


23 


13. 49 


10- 35 


13. 40 


10. 47 




18 


14. 47 


10. 


71 


14. 


38 


10. 


83 


14. 28 


10- 96 


14. 18 


11. 08 




19 


15. 27 


11. 


30 


15. 


17 


11. 


43 


15. 07 


11- 57 


14. 97 


11. 70 




20 


16. 08 


11. 


90 


15. 


97 


12. 


04 


15. 87 


12- 18 


15. 76 


12. 31 




21 


16. 88 


12. 


49 


16. 


77 


12. 


64 


16. 66 


12- 78 


16. 55 


12. 93 




22 


17. 68 


13. 


09 


17. 


57 


13. 


24 


17. 45 


13- 39 


17. 34 


13. 54 




23 


18. 49 


13. 


68 


18. 


37 


13. 


84 


18. 25 


14- 00 


18. 12 


14. 16 




24 


19. 29 


14. 


28 


19. 


17 


14. 


44 


19. 04 


14- 61 


18. 91 


14. 78 




25 


20. 10 


14. 


87 


19. 


97 


15. 


05 


19. 83 


15- 22 


19. 70 


15. 39 




26 


20. 90 


15. 


47 


20. 


76 


15. 


65 


20. 63 


15. 83 


20. 49 


16. 01 




27 


21. 70 


16. 


06 


21. 


56 


16. 


25 


21. 42 


16- 44 


21. 28 


16. 62 




28 


22. 51 


16. 


65 


22. 


36 


16. 


85 


22. 21 


17. 05 


22. 06 


17. 24 




29 


23. 31 


17. 


25 


23. 


16 


17. 


45 


23. 01 


17. 65 


22. 85 


17. 85 




30 


24. 12 


17. 


84 


23. 


96 


18. 


05 


23. 80 


18- 26 


23. 64 


18. 47 




35 


28. 13 


20. 


82 


27. 


95 


21. 


06 


27. 77 


21- 31 


27. 58 


21. 55 




40 


32. 15 


23. 


79 


31. 


95 


24. 


07 


31. 73 


24- 35 


31. 52 


24. 63 




45 


36. 17 


26. 


77 


35. 


94 


27. 


08 


35. 70 


27- 39 


35. 46 


27. 70 




50 


40. 19 


29. 


74 


39. 


93 


30. 


09 


39. 67 


30- 44 


39. 40 


30. 78 




55 


44. 21 


32. 


72 


43. 


92 


33. 


10 


43. 63 


33- 48 


43. 34 


33. 86 




60 


48. 23 


35. 


69 


47. 


92 


36. 


11 


47. 60 


36- 53 


47. 28 


36. 94 




65 


52. 25 


38. 


66 


51 


91 


39. 


121 


51.57 


39- 57 


51. 22 


40. 02 




70 


56. 27 


41. 


64 


55. 


90 


42. 


13 


55. 53 


42- 61 


55. 16 


43. 10 




75 


60. 29 


44. 


61 


59. 


90 


45. 


14 


59. 50 


45. 66 


59. 10 


46. 17 




80 


64. 31 


47. 


59 


63. 


89 


48. 


15 


63. 47 


48. 70 


63. 04 


49. 25 




85 


68. 33 


50. 


56 


67. 


48 


51. 


15 


67. 43 


51. 74 


66. 98 


52. 33 




90 


72. 35 


53. 


53 


71. 


88 


54. 


16 


71. 40 


54. 79 


70. 92 


55. 41 




95 


76. 37 


56. 


51 


75. 


87 


57. 


17 


75. 37 


57. 83 


74. 86 


58. 49 




100 


80. 39 


59. 


48 


79. 


86 


60. 


18 


79. 34 


60. 88 


78. 80 


61. 57 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




533^ 


Deg. 






531 


>eg. 




52^ Deg. 


52] 


)eg. 







90 


TRAVERSE TABLE. 








55' 


38% Deg. 


39 Deg. 


39% Deg. 


40 Deg. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




1 


0. 78 


0. 62 


0. 78 


0. 63 


0. 


77 


0. 64 


0. 77 


0. 64 




2 


1. 57 


1. 24 


1. 55 


1. 26 


1. 


54 


1.27 


1. 53 


1. 29 




3 


2. 35 


1. 87 


2. 33 


1. 89 


2. 


31 


1. 91 


2. 30 


1. 93 




4 


3. 13 


2. 49 


3. 11 


2. 52 


3. 


09 


2. 54 


3. 06 


2. 57 




5 


3. 91 


3. 11 


3. 89 


3. 15 


3. 


86 


3. 18 


3. 83 


3. 21 




6 


4. 70 


3. 74 


4. 66 


3. 98 


4. 


63 


3. 82 


4. 60 


3. 86 




7 


5. 48 


4. 36 


5. 44 


4. 41 


5. 


40 


4. 45 


5. 36 


4. 50 




8 


6. 26 


4. 98 


6. 22 


5. 03 


6. 


17 


5. 09 


6. 13 


5. 14 




9 


7. 04 


5. 60 


6. 99 


5. 66 


6. 


94 


5. 72 


6. 89 


5. 79 




10 


7. 83 


6. 23 


7. 77 


6„ 29 


7. 


72 


6. 36 


7. 66 


6. 43 




11 


8. 61 


6. 85 


8. 55 


6. 92 


8. 


49 


7. 00 


8. 43 


7. 07 




12 


9. 39 


7. 47 


9. 33 


7. 55 


9. 


26 


7. 63 


9. 19 


7. 71 




13 


10. 17 


8. 09 


10. 10 


8. 18 


10. 


03 


8. 27 


9. 96 


8. 36 




14 


10. 96 


8. 72 


10. 88 


8. 81 


10. 


80 


8. 91 


10. 72 


9. 00 




15 


11. 74 


9. 34 


11. 66 


9. 44 


11. 


57 


9. 54 


11. 49 


9. 64 




16 


12. 52 


9. 96 


12. 43 


10. 07 


12. 


35 


10. 18 


12. 26 


10. 28 




17 


13. 30 


10.58 


13. 21 


10. 70 


13. 


12 


10. 81 


13. 02 


10. 93 




18 


14. 09 


11.21 


13. 99 


11. 33 


13. 


89 


11. 45 


13. 79 


11. 57 




19 


14. 87 


11. 83 


14.77 


11. 96 


14. 


66 


12. 09 


14. 55 


12. 21 




i 20 


15. 65 


12. 45 


15. 54 


12. 59 


15. 


43 


12. 72 


15. 32 


12. 86 




21 


16. 43 


13. 07 


16. 32 


13. 22 


16. 


20 


13. 36 


16. 09 


13. 50 




22 


17. 22 


13. 70 


17. 10 


13. 84 


16. 


98 


13. 99 


16. 85 


14. 14 




23 


18. 00 


14. 32 


17. 87 


14. 47 


17. 


75 


14. 63 


17. 62 


14. 78 




24 


18. 78 


14. 94 


18. 65 


15. 10 


18. 


52 


15.27 


18. 39 


15. 43 




25 


19. 57 


15. 56 


19. 43 


15. 73 


19. 


29 


15. 90 


19. 15 


16. 07 




26 


20. 35 


16. 19 


20. 21 


16. 36 


20. 


06 


16. 54 


19. 92 


16. 71 




27 


21. 13 


16.81 


20. 98 


16. 99 


20. 


83 


17. 17 


20. 68 


17. 36 




! 28 


21.91 


17.43 


21. 76 


17. 62 


21. 


61 


17.81 


21. 45 


18. 00 




1 29 


22. 70 


18.05 


22. 54 


18. 25 


22. 


38 


18. 45 


22. 22 


18. 64 




30 


23. 48 


18. 68 


23. 31 


18. 88 


23. 


15 


19. 08 


22. 98 


19. 28 




35 


27. 39 


21. 79 


27. 20 


22. 03 


27. 


01 


22. 26 


26. 81 


22. 50 




40 


31. 30 


24. 90 


31. 09 


25. 17 


30. 


86 


25. 44 


30. 64 


25. 71 




45 


35. 22 


28. 01 


34. 97 


28. 32 


34. 


72 


28. 62 


34. 47 


28. 93 




50 


39. 13 


31. 13 


38. 86 


31. 47 


38. 


58 


31. 80 


38. 30 


32. 14 




55 


43. 04 


34. 24 


42. 74 


34. 61 


42. 


44 


34. 98 


42. 13 


35. 35 




60 


46. 96 


37. 35 


46. 63 


37. 76 


46. 


30 


38. 16 


45. 96 


38. 57 




65 


50. 87 


40. 46 


50. 51 


40. 91 


50. 


16 


41. 35 


49. 79 


41. 78 




70 


54. 78 


43. 58 


54. 40 


44. 05 


54. 


01 


44. 53 


53. 62 


45. 00 




75 


58. 70 


46. 69 


58. 29 


47. 20 


57. 


87 


47. 71 


57. 45 


48. 21 




80 


62. 61 


49. 80 


62. 17 


50. 35 


61. 


73 


50. 89 


61. 28 


51. 42 




85 


66. 52 


52. 91 


66. 06 


53. 49 


65. 


59 


54. 07 


65. 11 


54. 64 




90 


70. 43 


56. 03 


69. 94 


56. 64 


69. 


45 


57. 25 


68. 94 


57. 85 




95 


74. 35 


59. 14 


73. 83 


59. 79 


73. 


30 


60. 43 


72. 77 


61. 06 




100 


78. 26 


62.25 


77.71 


62. 93 


77. 


16 


63. 61 


76. 60 


64. 28 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




51% 


Deg. 


51 ] 


Deg. 


50% Deg. 


50 Deg. 









TRAVERSE TABLE. 91 


S3" 

1 
n 
Of 

1 


40^ Deg. 


41 Deg. 


41^ Deg. 


42 Deg. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


0.76 


0.65 


0. 75 


66 


0. 75 


0. 66 


0. 74 


0. 67 


2 


1.52 


1. 30 


1. 51 


1.31 


1. 50 


1. 33 


1.4^ 


1. 34 


3 


2.28 


1. 95 


2. 26 


1. 97 


2.25 


1. 99 


2. 23 


2. 01 


4 


3. 04 


2. 60 


3. 02 


2. 62 


3. 00 


2. 65 


2. 97 


2. 68 


5 


3. 80 


3. 25 


3.77 


3. 28 


3. 74 


3. 31 


3. 72 


3. 35 


6 


4.56 


3. 90 


4. 53 


3. 94 


4. 49 


3. 98 


4. 46 


4. 01 


7 


5. 32 


4. 55 


5. 28 


4. 59 


5. 24 


4- 64 


5. 20 


4. 68 


8 


6. 08 


5.20 


6. 04 


5.25 


5. 99 


5. 30 


5. 95 


5. 35 


9 


6. 84 


5.84 


6. 79 


5. 90 


6. 74 


5. 96 


6. 69 


6. 02 


10 


7. 60 


6. 49 


7.55 


6. 56 


7. 49 


6. 63 


7. 43 


6. 69 


11 


8. 36 


7. 14 


8. 30 


7. 22 


8. 24 


7.29 


8. 17 


7. 36 


12 


9. 12 


7. 79 


9. 06 


7. 87 


8. 99 


7. 95 


8. 92 


8. 03 


13 


9. 89 


8. 44 


9. 81 


8. 53 


9. 74 


8. 61 


9. 66 


8. 70 


14 


10. 65 


9. 09 


10. 57 


9. 18 


10. 49 


9. 28 


10. 40 


9. 37 


15 


11. 41 


9. 74 


11. 32 


9. 84 


11. 23 


9- 94 


11. 15 


10. 04 


16 


12. 17 


10. 39 


12. 08 


10. 50 


11. 98 


10- 60 


11. 89 


10. 71 


17 


12. 93 


11. 04 


12. 83 


11. 15 


12. 73 


11-26 


12. 63 


11. 38 


18 


13. 69 


.11.-69 


13. 58 


11. 81 


13. 48 


11. 93 


13. 38 


12. 04 


.19 


14. 45 


12. 34 


14. 34 


12. 47 


14. 23 


12.59 


14. 12 


12. 71 


20 


15. 21 


12. 99 


15. 09 


13. 12 


14. 98 


13. 25 


14. 86 


13. 38 


21 


15. 97 


13. 64 


15. 85 


13. 78 


15.73 


13- 91 


15. 61 


14. 05 


22 


16. 73 


14. 29 


16. 60 


14. 43 


16. 48 


14- 58 


16. 35 


14. 72 


23 


17.49 


14. 94 


17. 36 


15. 09 


17. 23 


15- 24 


17. 09 


15. 39 


24 


18. 25 


15. 59 


18. 11 


15. 75 


17. 97 


15. 90 


17. 84 


16. 06 


25 


19. 01 


16. 24 


18. 87 


16. 40 


18. 72 


16- 57 


18. 58 


16. 73 


26 


19. 77 


16. 89 


19. 62 


17. 06 


19. 47 


17. 23 


19. 32 


17. 40 


27 


20. 53 


17. 54 


20. 38 


17. 71 


20. 22 


17- 89 


20. 06 


18. 07 


.28 


21.29 


18. 18 


21. 13 


18. 37 


20. 97 


18- 55 


20. 81 


18. 74 


29 


22. 05 


18. 83 


21. 89 


19. 03 


21. 72 


19-22 


21. 55 


19. 40 


30 


22. 81 


19. 48 


22. 64 


19. 68 


22. 47 


19- 88 


22. 29 


20. 07 


35 


26. 61 


22. 73 


26. 41 


22. 96 


26. 21 


23- 19 


26. 01 


23. 42 


40 


30.42 


25. 98 


30. 19 


26. 24 


29. 96 


26- 50 


29. 73 


26. 77 


45 


34. 22 


29. 23 


33. 96 


29. 52 


33. 70 


29- 82 


33. 44 


30. 11 


50 


38. 02 


32. 47 


37. 74 


32. 80 


37. 45 


33- 13 


37. 16 


33. 46 


55 


41. 82 


35. 72 


41. 51 


36. 08 


41. 19 


36- 44 


40. 87 


36. 80 


60 


45. 62 


38. 97 


45. 28 


39.36 


44. 94 


39- 76 


44. 59 


40. 15 


65 


49. 43 


42. 21 


49. 06 


42. 64 


48. 68 


43- 07 


48. 30 


43. 49 


70 


53. 23 


45. 46 


52. 83 


45. 92 


52. 43 


46. 38 


52. 02 


46. 84 


75 


57. 03 


48. 71 


56. 60 


49. 20 


56. 17 


49. 70 


55. 74 


50. 18 


80 


60. 83 


51. 96 


60. 38 


52. 48 


59. 92 


53. 01 


59. 45 


53. 53 


85 


64. 63 


55. 20 


64. 15 


55. 76 


63. 66 


56. 32 


63. 17 


56. 88 


90 


68. 44 


58. 45 


67. 92 


59. 05 


67. 41 


59. 64 


66. 88 


60. 22 


95 


72. 24 


61. 70 


71. 70 


62. 33 


71. 15 


62. 95 


70. 60 


63.57 


100 


76. 04 


64. 98 


75. 47 


65. 61 


74. 90 


66.26 
Lat. 


74. 31 


66. 91 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Dep. 


Lat. 


49^ Deg. 


49 Deg. 


48^ Deg. 


48 Deg. I 





22 



92 TRAVERSE TABLE. 




I 


42% Deg. 


43 Deg. 


4314 Deg. 


44 Deg. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




l 


0. 74 


0. 68 


0. 73 


0. 68 


0. 73 


0. 69 


0. 72 


0. 69 




2 


1. 47 


1. 35 


1. 46 


1. 36 


1. 45 


1. 38 


1. 44 


1. 39 




3 


2. 21 


2. 03 


2. 19 


2. 05 


2. 18 


2.07 


2. 16 


2. 08 




4 


2. 95 


2.70 


2. 93 


2. 73 


2. 90 


2.75 


2. 88 


2. 78 




5 


3. 69 


3. 38 


3. 66 


3. 41 


3. 63 


3. 44 


3. 60 


3. 47 




6 


4. 42 


4. 05 


4. 39 


4. 09 


4. 35 


4. 13 


4. 32 


4. 17 




7 


5. 16 


4. 73 


5. 12 


4. 77 


5. 08 


4. 82 


5. 04 


4. 86 




8 


5. 90 


5. 40 


5. 85 


5. 46 


5.80 


5. 5l 


5.75 


5. 56 




9 


6. 64 


6. 08 


6. 58 


6. 14 


6. 53 


6.20 


6. 47 


6. 25 




10 


7. 37 


6.76 


7. 31 


6. 82 


7. 25 


6. 88 


7. 19 


6. 95 




11 


8. 11 


7. 43 


8. 04 


7. 50 


7. 98 


7. 57 


7. 91 


7. 64 




12 


8. 85 


8. 11 


8. 78 


8. 18 


8. 70 


8. 26 


8. 63 


8. 34 




13 


9. 58 


8. 78 


9. 51 


8. 87 


9. 43 


8.95 


9. 35 


9. 03 




14 


10. 32 


9. 46 


10. 24 


9. 55 


10. 16 


9. 64 


10. 07 


9. 73 




15 


11. 06 


10. 13 


10. 97 


10. 23 


10. 88 


10. 33 


10. 79 


10. 42 




16 


11. 80 


10. 81 


11. 70 


]0. 91 


11. 61 


11. 01 


11.51 


11. 11 




17 


12. 53 


11. 48 


12. 43 


11. 59 


12. 33 


11. 70 


12. 23 


11. 81 




18 


13. 27 


12. 16 


13. 16 


12. 28 


13. 06 


12. 39 


12. 95 


12.50 




19 


14. 01 


12. 84 


13. 90 


12. 96 


13. 78 


13. 08 


13. 67 


13. 20 




20 


14. 75 


13. 51 


14. 63 


13. 64 


14. 51 


13. 77 


14. 39 


13. 89 




21 


15. 48 


14. 19 


15. 36 


14. 32 


15. 23 


14. 46 


15. 11 


14. 59 




22 


16. 22 


14. 86 


16. 09 


15. 00 


15. 96 


15. 14 


15. 83 


15. 28 




23 


16.96 


15. 54 


16. 82 


15. 69 


16. 88 


15. 83 


16. 54 


15. 98 




24 


17. 69 


16. 21 


17. 55 


16. 37 


17.41 


16. 52 


17. 26 


16. 67 




25 


18. 43 


16. 89 


18. 28 


17. 05 


18. 13 


17.21 


17. 98 


17. 37 




26 


19. 17 


17.57 


19. 02 


17. 73 


18. 86 


17. 90 


18.70 


18. 06 




27 


19.91 


18.24 


19. 75 


18. 41 


19. 59 


18. 59 


19. 42 


18. 76 




28 


20. 64 


18. 92 


20. 48 


19.. 10 


20. 31 


19. 27 


20. 14 


19. 45 




29 


21. 38 


19.59 


21. 21 


19.78 


21. 04 


19. 96 


20. 86 


20. 15 




30 


22. 12 


20. 27 


21. 94 


20. 46 


21.76 


20. 65 


21.58 


20. 84 




35 


25. 80 


23. 65 


25. 60 


23. 87 


25. 39 


24. 09 


25. 18 


24. 31 




40 


29. 49 


27. 02 


29. 25 


27.28 


29.01 


27. 53 


28. 77 


27. 79 




45 


33. 18 


30. 40 


32. 91 


30. 69 


32. 64 


30. 98 


32. 37 


31. 26 




50 


36. 86 


33.78 


36. 57 


34. 10 


36.27 


34.42 


35. 57 


34. 73 




55 


40. 55 


37. 16 


40. 22 


37.51 


39. 90 


37. 86 


39. 96 


38. 21 




60 


44. 24 


40. 54 


43. 88 


40. 92 


43. 52 


41. 30 


43. 16 


41. 68 




65 


47. 92 


43.91 


47.54 


44. 33 


47. 15 


44. 74 


46. 76 


45. 15 




70 


51. 61 


47. 29 


51. 19 


47.74 


50. 78 


48. 18 


50. 35 


48. 63 




75 


55. 30 


50. 67 


54. 85 


51. 15 


54. 40 


51. 63 


53. 95 


52. 10 




80 


58. 98 


54. 05 


58. 51 


54. 56 


58. 03 


55. 07 


57. 55 


55.57 




85 


62. 67 


57.43 


62. 17 


57. 97 


61. 66 


58. 51 


61. 14 


59. 05 




90 


66. 35 


60. 80 


65. 82 


61. 38 


65.28 


61. 95 


64. 74 


62. 52 




95 


70. 04 


64. 18 


69. 48 


64. 79 


68. 91 


65. 39 


68. 34 


65. 99 




100 


73. 73 


67.56 


73. 14 


68.20 


72. 54 


68. 84 


71. 93 


69. 47 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




47J4 Deg. 


47 Deg. 


46% Deg. 


46 Deg. 









TRAVERSE TABLE. 


93 




2 
s* 

1 

1 


44^ Deg. 


45 Deg. 




Lat. 


Dep. 


Lat. 


Dep. 


0.71 


0. 70 


0. 71 


71 




2 


1. 43 


1. 40 


1. 41 


1.41 






3 


2. 14 


2. 10 


2. 12 


2. 12 






4 


2. 85 


2. 80 


2. 83 


2. 83 






5 


3. 57 


3. 50 


3. 54 


3. 54 






6 


4. 28 


4. 21 


4. 24 


4. 24 






7 


4. 99 


4. 91 


4. 95 


4. 95 






8 


5. 71 


5. 61 


5. 66 


5. 66 






9 


6. 42 


6. 31 


6. 36 


6. 36 






10 


7. 13 


7. 01 


7. 07 


7. 07 






11 


7. 85 


7. 71 


7. 78 


7. 78 






12 


8. 56 


8. 41 


8. 49 


8. 49 






13 


9.27 


9. 11 


9. 19 


9. 19 






14 


9. 99 


9. 81 


9. 90 


9. 90 






15 


10. 70 


10. 51 


10. 61 


10. 61 






16 


11. 41 


11. 21 


11. 31 


11. 31 






17 


12. 13 


11. 92 


12. 02 


12. 02 






18 


12. 84 


12. 62 


12. 73 


12. 73 






19 


13. 55 


13. 32 


13. 43 


13. 43 






20 


14. 26 


14. 02 


14. 14 


14. 14 






21 


14. 98 


14. 72 


14. 85 


14. 85 






22 


15. 69 


15. 42 


15. 56 


15.56 






23 


16. 40 


16. 12 


16. 26 


16. 26 






24 


17. 12 


16. 82 


16. 97 


16. 97 






25 


17. 83 


17. 52 


17. 68 


17. 68 






26 


18. 54 


18. 22 


18. 38 


18. 38 






27 


19.26 


18. 92 


19. 09 


19. 09 






28 


19. 97 


19. 63 


19. 80 


19. 80 






29 


20. 68 


20. 33 


20. 51 


20.51 






30 


21. 40 


21. 03 


21. 21 


21.21 






35 


24. 96 


24. 53 


24. 75 


24.75 






40 


28. 53 


28. 04 


28. 28 


28. 28 






45 


32. 10 


31. 54 


31. 82 


31. 82 






50 


35. 66 


35. 05 


35. 36 


35. 36 






55 


39. 23 


38. 55 


38. 89 


38. 89 






60 


42. 79 


42. 05 


42. 43 


42. 43 






65 


46. 36 


45. 56 


45. 96 


45. 96 






70 


49. 93 


49. 06 


49. 50 


49. 50 






75 


53. 49 


52. 57 


53. 03 


53. 03 






80 


57. 06 


56. 07 


56. 57 


56. 57 






85 


60. 63 


59. 58 


60. 10 


60. 10 






90 


64. 19 


63. 08 


63. 64 


63. 64 






95 


67. 76 


66. 59 


67. 18 


67. 18 






100 


71. 33 


70. 09 


70.71 


70.71 




Dep. 


Lat. 


Dep. 


Lat. 


45}^ Deg. 


45 Deg. 







94 


Meridianal Parts. TABLE IV. 




' 


0° 


1 l 2° 3 3 


4° 


5° 


6° 


7° 


8° 


9° 


10° 


11° 


12° 


13° 


14° 


15° 




~b 


~0 


60 120 


180 


240 


3oa 


361 


421 


482 


542 


603 


664 


725 


787 


8?8 


910 




1 


1 


61 


121 


181 


241 


301 


362 


422 


483 


643 


604 


665 


726 


788 


850 


911 




2 


3 


62 


122 


182 


242 


302 


363 


423 


484 


544 


605 


666 


727 


789 


851 


913 




3 


3 


63 


123 


183 


243 


303 


364 


424 


485 


545 


606 


667 


728 


790 


852 


914 




4 


4 


64 


124 


184 


244 


304 


365 


425 


486 


546 


607 


668 


729 


791 


853 


916 




6 


5 


65 


125 


185 


245 


305 


366 


426 


487 


547 


608 


669 


730 


792 


854 


916 




6 


6 


66 


126 


186 


246 


303 


367 


427 


488 


548 


609 


670 


731 


793 


855 


917 




7 


7 


67 


127 


187 


247 


307 


368 


428 


489 


549 


610 


671 


732 


794 


856 


918 




8 


8 


68 


128 


188 


248 


308 


369 


429 


490 


550 


611 


672 


734 


795 


867 


919 




9 


9 


69 


129 


189 


249 


309 


370 


430 


491 


551 


612 


673 


735 


796 


858 


920 




10 


10 


70 


130 


190 


250 


310 


371 


431 


492 


552 


613 


674 


736 


797 


859 


921 




11 


11 


71 


131 


191 


251 


311 


372 


432 


493 


553 


614 


675 


737 


798 


860 


922 




12 


12 


72 


132 


192 




























13 


13 


73 


133 


193 


253 


313 


374 


434 


495 


665 


616 


677 


739 


800 


862 


924 




14 


14 


74 


134 


194 


254 


314 


375 


435 


496 


556 


617 


678 


740 


801 


863 


925 




15 


15 


75 


135 


195 


255 


315 


376 


436 


497 


557 


618 


679 


741 


802 


864 


926 




16 


16 


76 


136 


196 


256 


316 


377 


437 


468 


558 


619 


680 


742 


803 


865 


927 




17 


17 


77 


137 


197 


257 


317 


378 


438 


499 


559 


620 


681 


743 


804 


366 


928 




18 


18 


78 


138 


198 


258 


318 


379 


439 


500 


560 


621 


682 


744 


805 


867 


929 




19 


19 


79 


139 


199 


259 


319 


380 


440 


501 


561 


622 


683 


745 


806 


868 


930 




20 


20 


80 


140 


200 


260 


320 


381 


441 


502 


562 


623 


684 


746 


807 


869 


931 




21 


21 


81 


141 


201 


261 


321 


382 


442 


503 


664 


624 


685 


747 


808 


870 


932 




22 


2-2 


82 


142 


202 


262 


322 


383 


443 


504 


565 


625 


687 


748 


809 


871 


933 




23 


23 


83 


143 


203 


263 


323 


384 


444 


505 


566 


626 


688 


749 


810 


872 


934 




24 


24 


84 


144 


204 


264 


824 


385 


445 


506 


667 


627 


689 


760 


811 


873 


935 




26 


25 


85 


145 


205 


265 


325 


386 


446 


507 


668 


628 


690 


751 


812 


874 


936 




26 


26 


86 


146 


206 


266 


326 


387 


447 


508 


569 


629 


691 


752 


313 


875 


937 




27 


27 


87 


147 


207 


267 


327 


388 


448 


509 


570 


631 


692 


763 


816 


876 


938 




28 


28 


88 


148 


208 


268 


328 


389 


449 


510 


671 


632 


693 


754 


816 


877 


939 




29 


29 


89 


149 


209 


269 


330 


390 


450 


511 


572 


633 


694 


755 


817 


878 


941 




30 


30 


90 


150 


210 


270 


331 


391 


451 


512 


573 


634 


695 


766 


818 


879 


942 




31 


31 


91 


151 


211 


271 


332 


392 


452 


613 


574 


655 


696 


767 


819 


880 


943 




32 


32 


92 


152 


212 


272 


333 


393 


463 


514 


575 


636 


697 


758 


820 


882 


944 




33 


33 


93 


153 


213 


273 


334 


394 


454 


516 


576 


637 


698 


769 


821 


883 


945 




34 


34 


94 


154 


214 


274 


335 


395 


455 


516 


577 


638 


699 


760 


822 


884 


946 




35 


35 


95 


155 


215 


275 


336 


396 


456 


517 


578 


639 


700 


761 


823 


885 


947 




36 


36 


96 


156 


216 


276 


337 


397 


457 


518 


579 


640 


701 


762 


824 


886 


948 




37 


37 


97 


157 


217 


277 


338 


398 


458 


519 


580 


941 


702 


763 


825 


887 


949 




38 


38 


98 


158 


218 


278 


339 


399 


459 


520 


581 


642 


703 


764 


826 


888 


950 




39 


39 


99 


159 


219 


279 


340 


400 


460 


521 


582 


643 


704 


765 


827 


889 


951 




40 


40 


100 


160 


220 


280 


341 


401 


461 


522 


583 


644 


705 


766 


828 


890 


952 




41 


41 


101 


161 


221 


381 


342 


402 


462 


523 


584 


645 


706 


767 


829 


891 


953 




42 


42 


102 


162 


222 


282 


343 


403 


463 


524 


585 


646 


707 


768 


830 


892 


954 




43 


43 


103 


163 


223 


283 


344 


404 


464 525 


586 


647 


708 


769 


831 


893 


955 




44 


44 


104 


164 


224 


284 


345 


405 


465 


526 


587 


648 


709 


770 


832 


894 


956 




45 


45 


105 


165 


225 


285 


346 


406 


466 


527 


588 


649 


710 


771 


833 


895 


957 




46 


46 


106 


166 


226 


286 


347 


407 


467 


528 


689 


650 


711 


772 


834 


896 


958 




47 


47 


107 


167 


227 


287 


348 


408 


468 


529 


690 


651 


712 


773 


836 


897 


959 




48 


48 


108 


168 


228 


288 


349 


409 


469 


530 


691 


652 


713 


774 


836 


898 


960 




49 


49 


109 


169 


229 


289 


350 


410 


470 


531 


692 


663 


714 


775 


837 


899 


961 




50 


50 


110 


170 


230 


290 


351 


411 


471 


532 


593 


654 


715 


777 


838 


900 


962 




51 


51 


111 


171 


231 


291 


352 


412 


472 


533 


694 


655 


716 


778 


839 


901 


963 




52 


52 


112 


172 


232 


292 


353 


413 


473 


534 


595 


656 


717 


779 


840 


902 


964 




53 


53 


113 


173 


233 


293 


364 


414 


474 


635 


596 


657 


718 


780 


841 


903 


965 




54 


54 


114 


174 


234 


294 


355 


415 


476 


536 


597 


658 


719 


781 


842 


904 


966 




55 


55 


115 


175 


235 


296 


366 


416 


477 


537 


598 


659 


720 


782 


843 


905 


968 




56 


56 


116 


176 


236 


296 


357 


417 


478 


538 


599 


660 


721 


783 


844 


606 


969 




57 


57 


117 


177 


237 


297 


358 


418 


479 


539 


600 


661 


722 


784 


846 


907 


970 




58 


5S 


118 


178 


238 


298 


359 


419 


480 


540 


601 


662 


723 


785 


846 


908 


971 




59 


59 


119 


179 


239 


299 


360 


420 


481 


541 


602 


663 


724 


786 847 


909 


972 











































TABLE IV. Meridianal Parts. 95 




/ 


16° 


17° 


18° 


19° 


20° 


21° 


22° 


23° 


24° 


25° 


26° 


27° 


28° 




~0 


"973 


1035 


1098 


1161 


1225 


1289 


1354 


1419 


1484 


1550 


16T6 


1684 


1761 




1 


974 


1036 


1099 


1163 


1226 


1290 


1355 


1420 


1485 


1551 


1618 


1685 


1752 




2 


975 


1037 


1100 


1164 


1227 


1291 


1356 


1421 


1486 


1552 


1619 


1686 


1753 




3 


976 


1038 


1101 


1165 


1228 


1292 


1357 


1422 


1487 


1553 


1620 


1687 


1755 




4 


977 


1039 


1102 


1166 


1229 


1293 


1358 


1423 


1488 


1554 


1621 


1688 


1756 




5 


978 


1041 


1103 


1167 


1230 


1295 


1359 


1424 


1490 


1556 


1622 


1689 


1757 




6 


979 


1042 


1105 


1168 


1232 


1296 


1360 


1425 


1491 


1557 


1623 


1690 


1758 




7 


980 


1043 


1106 


1169 


1233 


1297 


1361 


1426 


1492 


1558 


1624 


1692 


1759 




8 


981 


1044 


1107 


1170 


1234 


1298 


1362 


1427 


1493 


1559 


1625 


1693 


1760 




9 


982 


1045 


1108 


1171 


1235 


1299 


1363 


1428 


1494 


1560 


1626 


1694 


1761 




10 


983 


1046 


1109 


1172 


1236 


1300 


1364 


1430 


1495 


1561 


1628 


1695 


1762 




11 


984 


1047 


1110 


1173 


1237 


1301 


1366 


1431 


1496 


1562 


1629 


1696 


1764 




12 


985 


1048 


1111 


1174 


1238 


1302 


1367 


1432 


1497 


1563 


1630 


1697 


1765 




13 


986 


1049 


1112 


1175 


1239 


1303 


1368 


1433 


1498 


1564 


1631 


1698 


1766 




14 


987 


1050 


1113 


1176 


1240 


1304 


1369 


1434 


1499 


1565 


1632 


1699 


1767 




15 


988 


1051 


1114 


1177 


1241 


1305 


1370 


1435 


1500 


1567 


1633 


1700 


1768 




16 


989 


1052 


1115 


1178 


1242 


1306 


1371 


1436 


1502 


1568 


1634 


1701 


1769 




17 


990 


1053 


1116 


1179 


1243 


1307 


1372 


1437 


1503 


1569 


1635 


1703 


1770 




18 


991 


1054 


1117 


1181 


1244 


1308 


1373 


1438 


1504 


1570 


1637 


1704 


1772 




19 


993 


1055 


1118 


1182 


1245 


1310 


1374 


1439 


1505 


1571 


1638 


1705 


1773 




20 


994 


1056 


1119 


1183 


1246 


1311 


1375 


1440 


1506 


1572 


1639 


1706 


1774 




21 


995 


1057 


1120 


1184 


1248 


1312 


1376 


1441 


1507 


1573 


1640 


1707 


1775 




22 


996 


1058 


1121 


1185 


1249 


1313 


1377 


1443 


1508 


1574 


1641 


1708 


1776 




23 


997 


1059 


1122 


1186 


1250 


1314 


1379 


1444 


1509 


1575 


1642 


1709 


1777 




24 


998 


1060 


1123 


1187 


1251 


1315 


1380 


1445 


1510 


1577 


1643 


1711 


1778 




25 


999 


1161 


1125 


1188 


1252 


1316 


1381 


1446 


1511 


1578 


1644 


1712 


1780 




26 


1000 


1063 


1126 


1189 


1253 


1317 


1382 


1447 


1513 


1579 


1645 


1713 


1781 




27 


1001 


1064 


1127 


1190 


1254 


1318 


1383 


1448 


1514 


1580 


1647 


1714 


1782 




28 


1002 


1065 


1128 


1191 


1255 


1319 


1384 


1449 


1515 


1581 


1648 


1715 


1783 




29 


1003 


1066 


1129 


1192 


1256 


1320 


1385 


1450 


1516 


1582 


1649 


1716 


1784 




30 


1004 


1067 


1130 


1193 


1257 


1321 


1386 


1451 


1517 


1583 


1650 


1717 


1785 




31 


1005 


1068 


1131 


1194 


1258 


1322 


1387 


1452 


1518 


1584 


1651 


1718 


1786 




32 


1006 


1069 


1132 


1195 


1259 


1324 


1388 


1453 


1519 


1585 


1652 


1720 


1787 




33 


1007 


1070 


1133 


1196 


1260 


1325 


1389 


1455 


1520 


1586 


1653 


1721 


1789 




34 


1008 


1071 


1134 


1198 


1261 


1326 


1390 


1456 


1521 


1588 


1654 


1722 


1790 




35 


1009 


1072 


1135 


1199 


1262 


1327 


1392 


1457 


1522 


1589 


1656 


1723 


1791 




36 


1010 


1073 


1136 


1200 


1264 


1328 


1393 


1458 


1524 


1590 


1657 


1724 


1792 




37 


1011 


1074 


1137 


1201 


1265 


1329 


1394 


1459 


1525 


1591 


1658 


1725 


1793 




38 


1012 


1075 


1138 


1202 


1266 


1330 


1395 


1460 


1526 


1592 


1659 


1726 


1794 




39 


1013 


1076 


1139 


1203 


1267 


1331 


1396 


1461 


1527 


1593 


1660 


1727 


1795 




40 


1014 


1077 


1140 


1204 


1268 


1332 


1397 


1462 


1528 


1594 


1661 


1729 


1797 




41 


1015 


1078 


1141 


1205 


1269 


1333 


1398 


1463 


1529 


1595 


1662 


1730 


1798 




42 


1016 


1079 


1142 


1206 


1270 


1334 


1399 


1464 


1530 


1596 


1663 


1731 


1799 




43 


1018 


1080 


1144 


1207 


1271 


1335 


1400 


1465 


1531 


1598 


1664 


1732 


1800 




44 


1019 


1081 


1145 


1208 


1272 


1336 


1401 


1467 


1532 


1599 


1666 


1733 


1801 




45 


1020 


1082 


1146 


1209 


1273 


1338 


1402 


1468 


1533 


1600 


1667 


1734 


1802 




46 


1021 


1084 


1147 


1210 


1274 


1339 


1403 


1469 


1535 


1601 


1668 


1735 


1803 




47 


1022 


1085 


1148 


1211 


1275 


1340 


1405 


1470 


1536 


1602 


1669 


1736 


1805 




48 


1023 


1086 


1149 


1212 


1276 


1341 


1406 


1471 


1537 


1603 


1670 


1737 


1806 




49 


1024 


1087 


1150 


0213 


1277 


1342 


1407 


1472 


1538 


1604 


1671 


1739 


1807 




50 


1025 


1088 


1151 


1215 


1278 


1343 


1408 


1473 


1539 


1605 


1672 


1740 


1808 




61 


1026 


1089 


1152 


1216 


1280 


1344 


1409 


1474 


1540 


1606 


1673 


1741 


1809 




52 


1027 


1090 


1153 


1217 


1281 


1345 


1410 


1475 


1541 


1608 


1675 


1742 


1810 




53 


1028 


1091 


1154 


1218 


1282 


1346 


1411 


1476 


1542 


1609 


1676 


1743 


1811 




54 


1029 


1092 


1155 


1219 


1283 


1347 


1412 


1477 


1543 


1610 


1677 


1744 


1813 




55 


1030 


1093 


1156 


1220 


1284 


1348 


1413 


1479 


1544 


1611 


1678 


1746 


1814 




56 


1031 


1094 


1157 


1221 


1285 


1349 


1414 


1480 


1646 


1612 


1679 


1747 


1815 




57 


1032 


1095 


1158 


1222 


1286 


1350 


1415 


1481 


1547 


1613 


1680 


1748 


1816 




58 


1033 


1096 


1159 


1223 


1287 


1352 


1416 


1482 


1548 


1614 


1681 


1749 


1817 




59 


1034 


1097 


1160 


1224 


1288 


1353 


1418 


1483 


1549 


1615 


1682 


1750 


1818 











96 Meridianal Parts. TABLE IV. 


/ 


29° 


30° 


31° 


32° 


33° 


34° 


35° 


36° 


37° 


38° 


39° 


40° 


41° 


1 


1819 


1888 


1958 


2028 


2100 


2171 


2244 


23T8 


2393 


2468 


2545 


2623 


2702 


1 


1821 


1890 


1959 


2030 


2101 


2173 


2246 


2319 


2394 


2470 


2546 


2624 


2703 


2 


1822 


1891 


1960 


2031 


2102 


2174 


2247 


2320 


2395 


2471 


2648 


2625 


2704 


3 


1823 


1892 


1962 


2032 


2103 


2175 


2248 


2322 


2396 


2472 


2549 


2627 


2706 


4 


1924 


1893 


1963 


2033 


2104 


2176 


2249 


2323 


2398 


2473 


2550 


2628 


2707 


6 


1825 


1894 


1964 


2034 


2105 


2178 


2250 


2324 


2399 


2475 


2551 


2629 


2708 


6 


1826 


1895 


1965 


2035 


2107 


2179 


2252 


2325 


2400 


2479 


2553 


2631 


2710 


7 


1827 


1896 


1966 


2037 


2108 


2180 


2253 


2327 


2401 


2477 


2564 


2632 


2711 


8 


1829 


1898 


1967 


2038 


2109 


2181 


2254 


2328 


2403 


2478 


2556 


2633 


2712 


. 9 


1830 


1899 


1969 


2039 


2110 


2182 


2255 


2329 


2404 


2480 


2557 


2634 


2714 


10 


1831 


1900 


1970 


2040 


2111 


2184 


2257 


2330 


2405 


2481 


2558 


2636 


2715 


11 


1832 


1901 


1971 


2041 


2113 


2185 


2258 


2332 


2406 


2482 


2559 


2637 


2716 


12 


1833 


1902 


1972 


2043 


2114 


2186 


2269 


2333 


2408 


2484 


2660 


2638 


2718 


13 


1834 


1903 


1973 


2044 


2115 


2187 


2260 


2334 


2409 


2485 


2562 


2640 


2719 


14 


1835 


1905 


1974 


2045 


2116 


2188 


2261 


2335 


2410 


2486 


2563 


2641 


2720 


16 


1837 


1906 


1976 


2046 


2117 


2190 


2263 


2337 


2411 


2487 


2564 


2642 


2722 


16 


1838 


1907 


1977 


2047 


2119 


2191 


2264 


2338 


2413 


2489 


2566 


2644 


2723 


17 


1839 


1908 


1978 


2048 


2120 


2192 


2265 


2339 


2414 


2490 


2567 


2645 


2724 


18 


1840 


1909 


1679 


2050 


2121 


2193 


2266 


2340 


2415 


2491 


2568 


2646 


2726 


19 


1841 


1910 


1980 


2051 


2122 


2194 


2268 


2342 


2416 


2492 


2569 


2648 


2727 


20 


1842 


1912 


1981 


2052 


2123 


2196 


2269 


2343 


2418 


2494 


2571 


2649 


2728 


21 


1843 


1913 


1983 


2053 


2125 


2197 


2270 


2344 


2419 


2495 


2572 


2650 


2729 


22 


1845 


1914 


1984 


2054 


2126 


2198 


2271 


2345 


2420 


2496 


2573 


2551 


2731 


23 


1846 


1915 


1985 


2056 


2127 


2199 


2272 


2346 


2422 


2498 


2576 


2653 


2732 


24 


1847 


1916 


1986 


2057 


2128 


2200 


2274 


2348 


2423 


2499 


2576 


2654 


2733 


25 


1848 


1917 


1987 


2058 


2129 


2202 


2275 


2349 


2424 


2500 


2577 


2655 


2735 


29 


1849 


1918 


1988 


2059 


2131 


2203 


2266 


2350 


2425 


2501 


2578 


2657 


2736 


27 


1850 


1920 


1990 


2060 


2132 


2204 


2267 


2351 


2427 


2503 


2680 


2658 


2737 


28 


1852 


1921 


1991 


2061 


2133 


2205 


2279 


2353 


2428 


2504 


2581 


2659 


2739 


29 


1853 


1922 


1992 


2063 


2134 


2207 


2280 


2354 


2429 


2505 


2582 


2661 


2740 


30 


1854 


1923 


1993 


2064 


2135 


2208 


2281 


2355 


2430 


2506 


2584 


2662 


2742 


31 


1855 


1924 


1994 


2065 


2137 


2209 


2232 


2356 


2432 


2508 


2585 


2663 


2743 


32 


1856 


1925 


1995 


2066 


2138 


2210 


2283 


2358 


2433 


2509 


2586 


2665 


2744 


33 


1857 


1927 


1997 


2067 


2139 


2211 


2285 


2359 


2434 


2510 


2688 


2666 


2746 


34 


1858 


1928 


1998 


2069 


2140 


2213 


2286 


2360 


2435 


2512 


2589 


2667 


2747 


35 


1360 


1929 


1999 


2070 


2141 


2214 


2287 


2361 


2437 


2513 


2590 


2669 


2748 


36 


1861 


1930 


2000 


2071 


2143 


2215 


2288 


2363 


2438 


2514 


2591 


2670 


2750 


37 


1892 


1931 


2001 


2072 


2144 


2216 


2290 


2364 


2439 


2515 


2593 


2671 


2751 


38 


1863 


1932 


2002 


2073 


2145 


2217 


2291 


2365 


2440 


2517 


2594 


2673 


2752 


39 


1864 


1934 


2004 


2075 


2146 


2219 


2292 


2366 


2442 


2518 


2595 


2674 


2754 


40 


1865 


1935 


2005 


2076 


2147 


2220 


2293 


2368 


2443 


2519 


2597 


2675 


2765 


41 


1866 


1936 


2006 


2077 


2149 


2221 


2295 


2369 


2444 


2621 


2598 


2676 


2756 


42 


1868 


1937 


2077 


2078 


2150 


2222 


2296 


2370 


2445 


2522 


2599 


2678 


2758 


43 


1869 


1938 


2008 


2079 


2151 


2224 


2297 


2371 


2447 


2623 


2601 


2679 


2759 


44 


1870 


1939 


2010 


2080 


2152 


2225 


2298 


2373 


2448 


2524 


2602 


2680 


2760 


45 


1871 


1941 


2011 


2082 


2153 


2226 


2299 


2374 


2449 


2526 


2603 


2682 


2762 


46 


1872 


1942 


2012 


2083 


2155 


2227 


2301 


2375 


2451 


2527 


2604 


2683 


2763 


47 


1873 


1943 


2013 


2084 


2156 


2228 


2302 


2376 


2452 


2528 


2606 


2684 


2764 


48 


1875 


1944 


2014 


2085 


2157 


2230 


2303 


2378 


2453 


2530 


2907 


2636 


2766 


49 


1876 


1945 


2015 


2086 


2168 


2231 


2304 


2379 


2464 


2531 


2608 


2687 


2767 


50 


1877 


1946 


2017 


2088 


2159 


2232 


2306 


2380 


2456 


2532 


2610 


2688 


2768 


51 


1878 


1948 


2018 


2089 


2161 


2233 


2307 


2381 


2457 


2533 


2611 


2690 


2770 


52 


1879 


1949 


2019 


2090 


2162 


2235 


2308 


2383 


2458 


2535 


2612 


2691 


2771 


53 


1880 


1950 


2020 


2091 


2163 


2236 


2309 


2384 


2459 


2536 


2614 


2692 


2772 


54 


1881 


1951 


2021 


2092 


2164 


2237 


2311 


2386 


2461 


2537 


2615 


2694 


2774 


55 


1883 


1952 


2022 


2094 


2165 


2238 


2312 


2386 


2462 


2538 


2616 


2695 


2775 


56 


1884 


1953 


2024 


2095 


2167 


2239 


2313 


2388 


2463 


2540 


2617 


2696 


2776 


57 


1885 


1955 


2025 


2096 


2168 


2241 


2314 


2389 


2464 


2541 


2619 


2698 


2778 


68 


1886 


1956 


2026 


2097 


2169 


2242 


2316 


2390 


2466 


2542 


2620 


2699 


2779 


59 


1887 


1957 


2027 


2098 


2170 


2243 


2317 


2391 


2467 


2544 


2621 


2700 


2780 


I 



TABLE IV. Meridianal Parts. 97 




' 


42° 


43° 


440 


45° 


46° 


47° 


48° 


4y° 


50° 


51° 


52° 


5b° 


54° 




~0 


2782 


2863 


2946 


3030 


3lT6 


3203 


3292 


3382 


3474 


3~569 


3665 


3764 


3865 




1 


2783 


2864 


2947 


3031 


3117 


3204 


3293 


3384 


3476 


3570 


3667 


3765 


3866 




2 


2734 


2866 


2949 


3033 


3118 


3206 


3295 


3385 


3478 


3572 


3668 


3767 


3868 




3 


2786 


2867 


2950 


3034 


2120 


3207 


3296 


3387 


3479 


3573 


3670 


3769 


3870 




4 


2787 


2869 


2951 


3036 


3121 


3209 


3298 


3388 


3481 


3575 


3672 


3770 


3871 




6 


2788 


2870 


2953 


3037 


3123 


3210 


3299 


3390 


3482 


3577 


3673 


3772 


3873 




6 


2790 


2871 


2954 


3038 


3123 


3212 


3301 


3391 


3484 


3578 


3675 


3774 


3375 




7 


2791 


2873 


2956 


3040 


3126 


3213 


3302 


3393 


3485 


3580 


3677 


3776 


3877 




8 


2792 


2874 


2957 


3041 


3127 


3214 


3303 


3394 


3487 


3582 


3678 


3777 


3878 




9 


2794 


2875 


2958 


3043 


3129 


3216 


3305 


3396 


3488 


3583 


3680 


3779 


3880 




10 


2795 


2877 


2960 


3044 


3130 


3217 


3306 


3397 


3490 


3585 


3681 


378G 


3882 




11 


2797 


2878 


2961 


3046 


3131 


3219 


3308 


3399 


3492 


3586 


3683 


3782 


3883 




12 


2798 


2880 


2963 


3047 


3133 


3220 


3309 


3400 


3493 


3588 


3685 


3784 


3885 




13 


2799 


2881 


2964 


3048 


3134 


3222 


3311 


3402 


3495 


3590 


3686 


3785 


3887 




14 


2801 


2882 


2965 


3050 


3136 


3224 


3312 


3403 


3496 


3591 


3688 


3787 


3889 




16 


2802 


2884 


2967 


3051 


3137 


3225 


33141 3405 


3498 


3593 


3690 


3799 


3890 




16 


2803 


2885 


2968 


3053 


3139 


3226 


3316 


3407 


3499 


3594 


3691 


379C 


3892 




17 


2805 


2886 


2970 


3054 


3140 


3228 


3317 


3408 


3501 


3596 


3693 


3792 


3894 




18 


2806 


2888 


2971 


3055 


3142 


3229 


3319 


3410 


3503 


3598 


3695 


3794 


3895 




19 


2807 


2889 


2972 


3057 


3143 


3231 


3320 


3411 


3504 


3599 


3696 


3796 


3897 




20 


2809 


2891 


2974 


3058 


3144 


3232 


3322 


3413 


3506 


3601 


3698 


3797 


3899 




21 


2810 


2892 


2975 


3060 


3146 


3234 


3323 


3414 


3507 


3602 


3699 


3799 


3901 




22 


2811 


2893 


2976 


3061 


3147 


3235 


3325 


3416 


3509 


3604 


3701 


380C 


3902 




23 


2813 


2895 


2978 


3063 


3149 


3237 


3326 


3417 


3510 


3606 


3703 


3802 


3904 




24 


2814 


2896 


2979 


3084 


3150 


3238 


3328 


3419 


3512 


3607 


3704 


3804 


3906 




25 


2815 


2897 


2981 


3065 


3152 


3240 


3329 


3420 


3514 


3609 


3706 


380b 


3907 




26 


2817 


2899 


2982 


3067 


3153 


3241 


3331 


3422 


3515 


3610 


3708 


380? 


3909 




27 


2818 


2900 


2983 


3068 


3155 


3242 


3332 


3423 


3517 


3612 


3709 


3809 


3911 




28 


2820 


2902 


2985 


3070 


3156 


3244 


3334 


3425 


3518 


3614 


3711 


3811 


3913 




29 


2821 


2903 


2986 


3071 


3157 


3245 


3335 


3427 


3520 


3615 


3713 


3812 


3914 




30 


2822 


2904 


2988 


3073 


3159 


3247 


3337 


3428 


3521 


3617 


3714 


3814 


3916 




31 


2824 


2906 


2989 


3074 


3160 


3248 


3338 


3430 


3523 


3618 


3716 


3816 


3918 




32 


2825 


2907 


2991 


3075 


3162 


3250 


3340 


3431 


3525 


3620 


3717 


3817 


3919 




33 


2826 


2908 


2992 


3077 


3163 


3251 


3341 


3433 


3526 


3622 


3719 


3819 


3921 




34 


2828 


2910 


2993 


3078 


3165 


3253 


3343 


3434 


3528 


3623 


3721 


3821 


3923 




35 


2829 


2911 


2995 


3080 


3166 


3254 


3344 


3436 


3529 


3625 


3722 


3822 


3925 




36 


2830 


2913 


2996 


3081 


3168 


3256 


3346 


3437 


3531 


3626 


3724 


3824 


3926 




37 


2832 


2914 


2998 


3083 


3169 


3257 


3347 


3439 


3532 


3628 


3726 


3826 


3928 




38 


2833 


2915 


2999 


3084 


3171 


3259 


3349 


3440 


3534 


3630 


3727 


3827 


3930 




39 


2834 


2917 


3000 


3085 


3172 


3260 


3350 


3442 


3536 


3631 


3729 


3829 


3932 




40 


2836 


2918 


3002 


3087 


3173 


3262 


3352 


3543 


3537 


3633 


3731 


3831 


3933 




41 


2837 


2919 


3003 


3088 


3175 


3263 


3353 


3445 


3539 


3634 


3732 


3832 


3935 




42 


2839 


2921 


3005 


3090 


3176 


3265 


3355 


3447 


3540 


3636 


3734 


3834 


3937 




43 


2840 


2922 


2006 


3091 


3178 


3266 


3356 


3448 


3542 


3638 


3736 


3836 


3938 




44 


2841 


2924 


3007 


3093 


3179 


3268 


3358 


3450 


3543 


3639 


3737 


3838 


3940 




45 


2843 


2925 


3009 


3094 


3181 


3269 


3359 


3551 


3545 


3641 


3739 


3839 


3942 




46 


2844 


2926 


3010 


3095 


3182 


3271 


3361 


3453 


3547 


3643 


3741 


3841 


3944 




47 


2845 


2928 


3012 


3097 


3184 


3272 


3362 


3454 


3548 


3644 


3742 


3843 


3945 




48 


2847 


2929 


3013 


3098 


3186 


3274 


3364 


3456 


3550 


3646 


3744 


3844 


3947 




49 


2848 


2931 


3014 


3100 


3187 


3275 


3365 


3457 


3551 


3647 


3746 


3846 


3949 




50 


2849 


2932 


3016 


3101 


3188 


3277 


3367 


3459 


3553 


3649 


3747 


3848 


3951 




51 


2851 


2933 


3017 


3103 


3190 


3278 


3368 


3460 


3555 


3651 


3749 


3849 


3952 




52 


2852 


2935 


3019 


3104 


3191 


3280 


3370 


3462 


3556 


3652 


3750 


3851 


3954 




53 


2854 


2936 


3020 


3105 


3192 


3281 


3371 


3464 


3558 


3654 


3752 


3853 


3956 




54! 2855 


2937 


3021 


3107 


3194 


3283 


3373 


3465 


3559 


3655 


3754 


3864 


3958 




55 2856 


2939 


3023 


3108 


3195 


3284 


3374 


3467 


3561 


3657 


3755 


3856 


3959 




56 2858 


2940 


3024 


3110 


3197 


3286 


3376 


3468 


3562 


3659 


3757 


3858 


3961 




57 2859 


2942 


3026 


3111 


3198 


3287 


3378 


3470 


3564 


3660 


3759 


3860 


3963 




58 2860 


2943 


3027 


3113 


3200 


3289 


3379 


3471 


3566 


3662 


3760 


3861 


3964 




59 ' 2862 


2944 


3029 


3114 


3201 


3290 


3381 


3473 


3567 


3664 


3762 


3863 3966 









98 Meridianal Parts. TABLE IV. 


/ 


55° 


56° 


57° 


58° 


59° 


60° 


6P 


62° 


63° 


64° 


65° 


60° 


67° 


"o 


3968 


4074 


4183 


4294 


4409 


452/ 


4649 


4775 


4905 


5039 


5179 


5~324 


5l74 


1 


3970 


4076 


4184 


4296 


4411 


4529 


4651 


4777 


4907 


5042 


5181 


5326 


5477 


2 


3971 


4077 


4186 


4298 


4413 


4531 


4653 


4779 


4909 


5044 


5184 


5328 


5479 


3 


3973 


4079 


4188 


4300 


4415 


4533 


4665 


4781 


4912 


5046 


5186 


5331 


5482 


4 


3976 


4081 


4190 


4302 


4417 


4535 


4657 


4784 


4914 


5049 


5188 


5333 


5484 


5 


3977 


4083 


4192 


4304 


4419 


4537 


4660 


4786 


4916 


5051 


5191 


5336 


5487 


6 


3978 


4085 


4194 


4306 


4421 


4539 


4662 


4788 


4918 


5053 


5193 


5338 


5489 


7 


3980 


4086 


4195 


4308 


4423 


4541 


4664 


4790 


4920 


5055 


5195 


5341 


6492 


8 


3982 


4058 


4197 


4309 


4425 


4543 


4666 


4792 


4923 


5058 


5198 


6343 


5495 


9 


3984 


4080 


4199 


4311 


4427 


4545 


4668 


4794 


4925 


5060 


5200 


5346 


5497 


10 


3985 


4092 


4202 


4313 


4429 


4547 


4670 


4796 


4927 


5062 


5203 


5348 


5500 


11 


3987 


4094 


4203 


4315 


4431 


4549 


4672 


4798 


4929 


5065 


5205 


5351 


5502 


12 


3989 


4095 


4205 


4317 


4433 


4551 


4674 


4801 


4931 


5067 


5207 


5353 


5505 


13 


3991 


4097 


4207 


4319 


4434 


4553 


4676 


4805 


4934 


5069 


5210 


5356 


5507 


14 


3992 


4099 


4208 


4321 


4436 


4555 


4678 


4808 


4936 


5071 


5212 


6358 


5510 


15 


3994 


4101 


4210 


4323 


4438 


4557 


4680 


4807 


4938 


5074 


5214 


5361 


5513 


16 


3996 


4103 


4212 


4325 


4440 


4559 


4682 


4809 


4940 


5076 


5217 


5363 


5515 


17 


3998 


4104 


4214 


4327 


4442 


4562 


4684 


4811 


4943 


5078 


5219 


5366 


5518 


18 


3999 


4106 


4216 


4328 


4444 


4564 


4687 


4814 


4945 


5081 


5222 


5368 


5520 


19 


4001 


4108 


4218 


4330 


4446 


4566 


4689 


4816 


4947 


5083 


5224 


5371 


5523 


20 


4003 


4110 


4220 


4332 


4448 


4568 


4691 


4818 


4949 


5085 


5226 


5373 


5526 


21 


4005 


4112 


4221 


4334 


4450 


4570 


4693 


4820 


4951 


5088 


5229 


5376 


5528 


22 


4006 


4113 


4223 


4336 


4452 


4572 


4695 


4822 


4954 


5090 


5231 


5378 


5531 


23 


4008 


4115 


4225 


4338 


4454 


4574 


4697 


4824 


4956 


5092 


5234 


5380 


5533 


24 


4010 


4117 


4227 


4340 


4456 


4576 


4699 


4826 


4958 


5095 


5236 


5383 


5536 


25 


4012 


4119 


4229 


4342 


4458 


4578 


4701 


4829 


4960 


5097 


5238 


5385 


5539 


26 


4014 


4121 


4231 


4344 


4460 


4580 


4703 


4831 


4963 


5099 


5241 


5388 


5541 


27 


4015 


4122 


4232 


4346 


4462 


4582 


4705 


4833 


4965 


5102 


5243 


5390 


5544 


28 


4017 


4124 


4234 


4347 


4464 


4584 


4707 


4835 


4967 


5104 


5246 


5393 


5546 


29 


4019 


4126 


4236 


4349 


4466 


4586 


4710 


4837 


4969 


5106 


5248 


5395 


5549 


30 


4021 


4128 


4238 


4351 


4468 


4588 


4712 


4839 


4972 


5108 


5250 


5398 


5552 


31 


4022 


4130 


4240 


4353 


4470 


4590 


4714 


4842 


4974 


5111 


5253 


5401 


5554 


32 


4024 


4132 


4242 


4355 


4472 


4592 


4716 


4844 


4976 


5113 


5255 


5403 


5557 


33 


4026 


4133 


4244 


4367 


4474 


4594 


4718 


4846 


4978 


5115 


5258 


5406 


5559 


34 


4028 


4135 


4246 


4359 


4476 


4596 


4720 


4848 


4981 


5118 


5260 


5408 


5562 


35 


4029 


4137 


4247 


4361 


4478 


4598 


4722 


4850 


4983 


5120 


5263 


5411 


5565 


36 


4031 


4139 


4249 


4363 


4480 


4600 


4724 


4852 


4985 


5122 


6265 


5413 


5567 


37 


4033 


4141 


4251 


4365 


4482 


4602 


4726 


4855 


4987 


5126 


5267 


5416 


5570 


38 


4035 


4142 


4253 


4367 


4484 


4604 


4728 


4857 


4990 


5127 


5270 


6418 


5573 


39 


4037 


4144 


4255 


4369 


4486 


4606 


4731 


4859 


4992 


5129 


5272 


5421 


5575 


40 


4038 


4146 


4257 


4370 


4488 


4608 


4733 


4861 


4994 


5132 


5275 


5423 


5578 


41 


4040 


4148 


4259 


4372 


4490 


4610 


4735 


4863 


4996 


5134 


5277 


5426 


5580 


42 


4042 


4150 


4250 


4374 


4492 


4612 


4736 


4865 


4999 


5136 


5280 


5428 


5583 


43 


4044 


4152 


4262 


4376 


4494 


4614 


4739 


4868 


5001 


5139 


5282 


5431 


5586 


44 


4045 


4153 


4264 


4378 


4495 


4616 


4741 


4870 


5003 


5141 


5284 


5433 


5588 


45 


4047 


4155 


4266 


4380 


4497 


4618 


4743 


4872 


5005 


5143 


5287 


5436 


5591 


46 


4049 


4167 


4268 


4382 


4499 


4620 


4745 


4874 


5008 


5146 


5289 


5438 


6594 


47 


4051 


4159 


4270 


4384 


4501 


4623 


4747 


4876 


5010 


5148 


5292 


5441 


5596 


48 


4052 


4161 


4272 


4386 


4503 


4625 


4750 


4879 


5012 


5151 


5294 


5443 


5599 


49 


4054 


4162 


4274 


4388 


4505 


4627 


4752 


4881 


5014 


5153 


5297 


5446 


5602 


50 


4056 


4164 


4275 


4390 


4507 


4629 


4754 


4883 


5017 


5155 


5299 


5448 


5604 


51 


4058 


4166 


4277 


4392 


4509 


4631 


4756 


4885 


5019 


5158 


5301 


5451 


5607 


52 


4060 


4168 


4279 


4394 


4511 


4633 


4758 


4887 


5021 


5160 


5304 


5454 


5610 


53 


4061 


4170 


4281 


4396 


4513 


4635 


4760 


4890 


5023 


5162 


5306 


5456 


5612 


54 


4063 


4172 


4283 


4398 


4515 


4637 


4762 


4892 


5026 


5165 


5309 


5458 


6616 


55 


4065 


4173 


4285 


4399 


4517 


4639 


4764 


4894 


5028 


5167 


5311 


5461 


5617 


56 


4067 


4175 


4287 


4401 


4519 


4641 


4766 


4896 


5030 


5169 


5314 


5464 


5620 


57 


4069 


4177 


4289 


4403 


4521 


4643 


4769 


4898 


5033 


5172 


5316 


6466 


5623 


58 


4070 


4179 


4291 


4405 


4523 


4645 


4771 


4901 


5035 


5174 


5319 


5469 


5625 


59 


4072 


4181 


4292 


4407 


4525 


4647 


4773 


4903 


5037 


5176 


5321 


5471 


5628 

































TABLE IV. Meridianal Parts. 


99 




' 


68° 


69° 


70° 


71° 


72° 


73° 


74° 


75° 


76° 


77° 


78° 


7y° 


80° 




"o 


5631 


5795 


5966 


6146 


6335 


6534 


6746 


6970 


7210 


7467 


7745 


8046 


8375 




1 


5633 


5797 


5969 


6149 


6338 


6538 


6749 


6974 


7214 


7472 


7749 


8051 


8381 




2 


5636 


5800 


5972 


6152 


6341 


6541 


6753 


6978 


7218 


7476 


7754 


8056 


8387 




3 


5639 


5803 


5975 


6155 


6345 


6545 


6757 


6982 


7222 


7481 


7759 


8061 


8393 




4 


5642 


5806 


5978 


6158 


6348 


6548 


6760 


6986 


7227 


7485 


7764 


8067 


8398 




5 


5644 


5809 


5981 


6161 


6351 


6552 


6764 


6990 


7231 


7490 


7769 


8072 


8404 




6 


5646 


5811 


5984 


6164 


6354 


6555 


6768 


6994 


7235 


7494 


7774 


8077 


8410 




7 


5650 


5814 


6986 


6167 


6358 


6558 


6771 


6997 


7239 


7498 


7778 


8083 


8416 




8 


5652 


5817 


5989 


6170 


6361 


6562 


6775 


7001 


7243 


7503 


7783 


8088 


8422 




9 


5655 


5820 


5992 


6173 


6364 


6565 


6779 


7005 


7247 


7507 


7788 


8093 


8427 




10 


5658 


5823 


5995 


6177 


6367 


6569 


6782 


7009 


7252 


7512 


7793 


8099 


8433 




11 


5660 


5825 


5998 


6180 


6371 


6572 


6786 


7013 


7256 


7516 


7798 


8104 


8439 




12 


5663 


5828 


6001 


6183 


6374 


6576 


6790 


7017 


7260 


7521 


7803 


8109 


8445 




13 


5666 


5831 


6004 


6186 


6377 


6579 


6793 


7021 


7264 


7525 


7808 


8115 


8451 




14 


5668 


5834 


6007 


6189 


6380 


9583 


6797 


7025 


7268 


7530 


7813 


8120 


8457 




16 


5671 


5837 


6010 


6192 


6384 


6586 


6801 


7029 


7273 


7535 


7817 


8125 


8463 




16 


5674 


6839 


6013 


6195 


6387 


6590 


6804 


7033 


7277 


7439 


7821 


8131 


8469 




17 


5676 


6842 


6016 


6198 


6390 


6593 


6808 


7027 


7281 


7544 


7827 


8136 


8474 




18 


5679 


5845 


6019 


6201 


6394 


6597 


6812 


7041 


7285 


7548 


7832 


8141 


8480 




19 


5682 


5848 


6022 


6205 


6397 


6600 


6815 


7045 


7289 


7553 


7837 


8147 


8486 




20 


5685 


5851 


6025 


6208 


6400 


6603 


6819 


7048 


7294 


7557 


7842 


8152 


8492 




21 


5687 


5854 


6028 


6211 


6403 


5607 


6823 


7052 


7298 


7562 


7847 


8158 


8498 




22 


5690 


5856 


6031 


6214 


6407 


6610 


6826 


7056 


7302 


7566 


7852 


8163 


8504 




23 


5693 


5859 


6034 


6217 


6410 


6614 


6830 


7060 


7306 


7571 


7857 


8168 


8510 




24 


5695 


5862 


6037 


6220 


6413 


6617 


6834 


7064 


7311 


7576 


7862 


8174 


8516 




25 


5698 


5865 


6040 


6223 


6417 


6621 


6838 


7068 


7315 


7580 


7867 


9179 


8522 




26 


5701 


5868 


6043 


6226 


6420 


6624 


6841 


7072 


7319 


7585 


7872 


8185 


8528 




27 


5704 


5871 


6046 


6230 


6423 


6628 


6845 


7076 


7323 


7589 


7877 


8190 


8534 




28 


5706 


5874 


6049 


6233 


6427 


6631 


6849 


7080 


7328 


7594 


7882 


8196 


8540 




29 


5709 


5876 


6052 


6236 


6430 


6635 


6853 


7084 


7332 


7599 


7887 


8201 


8546 




30 


5712 


5879 


6055 


6239 


6433 


6639 


6856 


7088 


7336 


7603 


7892 


8207 


8552 




31 


5715 


5882 


6058 


6242 


6437 


6642 


6860 


7092 


7341 


7608 


7897 


8212 


8558 




32 


5717 


5885 


6061 


6245 


6440 


6646 


6864 


7096 


7345 


7612 


7902 


8218 


8565 




33 


5720 


5888 


6064 


6249 


6443 


6649 


6868 


7100 


7349 


7617 


7907 


8223 


8571 




34 


5723 


5891 


6067 


6252 


6447 


6653 


6871 


7104 


7353 


7622 


7912 


8229 


8577 




35 


5725 


5894 


6070 


6255 


6450 


6656 


6875 


7108 


7358 


7626 


7917 


8234 


8583 




36 


5728 


5896 


6073 


6258 


6453 


6660 


6879 


7112 


7362 


7631 


7922 


8240 


8589 




3? 


5731 


5899 


6076 


6261 


6457 


6663 


6883 


7116 


7366 


7636 


7927 


8245 


8595 




38 


5734 


5902 


6079 


6264 


6460 


6667 


6886 


7128 


7371 


7640 


7932 


8251 


8601 




39 


5736 


5905 


6082 


6268 


6463 


6670 


6890 


7124 


7375 


7645 


7937 


8256 


8607 




40 


5739 


5908 


6085 


6271 


6467 


6674 


6894 


7128 


7379 


7650 


7942 


8262 


8614 




41 


5742 


5911 


6088 


6274 


6470 


6677 


6898 


7132 


7384 


7654 


7948 


8267 


8620 




42 


5745 


6914 


6091 


6277 


6473 


6681 


6901 


7136 


7388 


7659 


7953 


8273 


8626 




43 


5747 


5917 


6094 


6280 


6477 


6685 


6905 


7140 


7392 


7664 


7958 


8279 


8632 




44 


5750 


5919 


6097 


6283 


6480 


6688 


6909 


7145 


7397 


7668 


7963 


8284 


8638 




45 


5753 


5922 


6100 


6287 


6483 


6692 


6913 


7149 


7401 


7673 


7968 


8290 


8644 




46 


5756 


5926 


6103 


6290 


6487 


6695 


6917 


7153 


7406 


7678 


7973 


8295 


8651 




47 


5768 


5928 


6106 


9293 


6490 


6699 


6920 


7157 


7410 


7683 


7978 


8301 


8657 




48 


5761 


6931 


6109 


6296 


6494 


6702 


6924 


7161 


7414 


7687 


7983 


8307 


8663 




49 


5764 


5934 


6112 


6299 


6497 


6706 


6928 


7165 


7419 


7692 


7989 


8312 


8669 




50 


5767 


5937 


6115 


6303 


6500 


6710 


6932 


7169 


7423 


7697 


7994 


8318 


8676 




51 


5770 


5940 


6118 


6306 


6504 


6713 


6936 


7173 


7427 


7702 


7999 


8324 


8682 




52 


6772 


5943 


6121 


6309 


6507 


6717 


6940 


7177 


7432 


7706 


8004 


8329 


8688 




53 


6775 


5946 


6124 


6312 


6611 


6720 


6943 


7181 


7436 


7711 


8009 


8336 


8695 




54 


5778 


5948 


6127 


6315 


6514 


6724 


6947 


7185 


7441 


7716 


8914 


8341 


8701 




55 


5781 


5951 


6130 


6319 


6517 


6728 


6951 


7189 


7445 


7721 


8020 


8347 


8707 




56 


5783 


5954 


6133 


6322 


6521 


6731 


6955 


7194 


7449 


7725 


8025 


8352 


8714 




57 


5786 


5957 


6136 


6325 


6524 


6735 


6959 


7198 


7464 


7730 


8030 


8358 


8720 




58 


5789 


5960 


6140 


6328 


6528 


6738 


6963 


7202 


7458 


7735 


8035 


8364 


8726 




59 


5792 


5963 


6143 


6332' 6531 


6742 


6966 


7206 


7463 


7740 


8040 


8369 


8733 




i 





100 



Meridianal Parts. 



TABLE IV. 



81° 82° 



8739 
8745 
8752 
8758 
8765 
8771 
8778 
8784 
8791 
8797 
8804 

11 8810 

12 8817 

13 8823 



14 
15 
16 
17 

18 
19 
20 

21 

22 
23 
24 
25 
26 
27 
28 
29 
30 

31 

32 
33 
34 
35 
36 
37 
38 
39 
40 

41 

42 
43 
44 
45 
46 
47 
48 
49 
50 

51 

52 
53 
54 
55 
56 
5? 
58 
59 



8830 
8836 
8843 
8849 
8856 
8863 



8876 
8883 



8896 



8909 
8916 
8923 
8930 
8936 

8943 
8950 
8957 
8963 
8970 
8977 



8991 
8998 
9005 

9012 
9018 
9025 
9032 
9039 
9046 
9053 
9060 
9067 
9074 

9081 
9088 
9096 
9103 
9110 
9117 
9124 
9131 
9138 



9145 
9153 
9160 
9167 
9174 
9182 
9189 
9197 
9203 
9211 
9218 

9225 
9233 
9240 
9248 
9255 
9262 
9270 
9277 
9285 
9292 

9300 
9307 
9315 
9322 
9330 
9337 
9345 
9353 
9360 
9368 

9376 
9383 
9391 
9399 
9407 
9414 
9422 
9430 
9438 
9445 

9453 
9461 
9469 
9477 
9485 
9493 
9501 
9509 
9517 
9525 

9533 
9541 
9549 
9557 
9565 
9573 
9581 
9589 
9598 



88° 



84° 



9606 
9614 
9622 
9631 
9639 
9647 
9655 
9664 
9672 
9683 
9689 

9697 
9706 
9714 
9723 
9731 
9740 
9748 
9757 
9765 
9774 

9783 
9791 
9800 
9809 
9817 
9826 
9835 
9844 
9852 
9861 

9870 
9879 



9906 
9915 
9924 
9933 
9942 
9951 

9960 

9969 

9978 

9987 

9996 

10005 

10015 

10024 

10033 

10043 

10052 
10061 
10071 
10080 
10089 
10099 
10108 
10118 
10127 



10137 
10146 
10156 
10166 
10175 
10185 
10195 
10205 
10214 
10224 
10234 

10244 
10254 
10364 
10273 
10283 
10293 
10303 
10314 
10324 
10334 

10344 
10354 
10364 
10374 
10385 
10395 
10405 
10416 
10426 
10437 

10447 
10457 
10468 
10479 
10489 
10500 
10510 
10521 
10532 
10542 

10553 
10564 
10575 
10586 
10597 
10608 
10619 
10630 
10641 
10652 

10663 
10674 
10685 
10696 
10708 
10719 
10730 
10742 
10753 



85° 



10765 
10776 
10788 
10799 
10811 
10822 
10834 
10846 
10858 
10869 
10881 



10905 
10917 
10929 
10941 
10953 
10965 
10978 
10990 
11002 

11014 
11027 
11039 
11052 
11064 
11077 
11089 
11102 
11115 
11127 

11140 
11153 
11166 
11179 
11192 
11205 
11218 
11231 
11244 
11257 

11270 
11284 
11297 
11310 
11324 
11337 
11351 
11365 
11378 
11392 

11406 
11420 
11434 
11448 
11462 
11476 
11490 
11504 
11518 



TABLE Y. 


TABLE VII. 


101 




Dip of the Sea Horizon. 


Mean Refraction of Celestial Objects. 




$£ 


?! 




gfi 


Alt 


. Refr 


Alt 


. Refr 


. Alt 


. Refr 


Alt 


Refr. 


1 Alt. 


Refr 




5"g 


o ~ 
2. o 


2« 


OT3 

2. o 


o 


/ / 


/ o 


/ / // o 


i / / 


o 


/ / // 


o 


ft 




2 ^ 


»s 


5" 2, 

2 S- 


33 < 


) 10 c 


>5 IE 


> 20 ( 


)2 35 


32 1 30 


67 


24 




222, 




S3, 


»t? 


10 31 3$ 


I H 


► 5 10 1( 


)2 24 


401 29 


68 


23 






"7 7/ 







20 29 5( 


) 2C 


5 05 


2( 


)2 22 


33 1 28 


69 


22 




1 


59 


38 
41 

44 


6 4 
6 18 
6 32 
6 45 

6 58 

7 10 
7 12 


30 28 21 


5 30 


5 0C 


► 3( 


)2 21 


201 26 


70 


21 




2 


I 24 


40 27 0( 


) 40 


4 56 


4( 


)2 29 


401 25 


71 


19 




3 


1 42 


50 25 42 


50 


4 51 


5( 


)2 28 


34 1 24 


72 


18 




4 


1 58 


47 


1 ( 


)24 2£ 


11 


4 47 


21 ( 


)2 27 


201 23 


73 


17 




5 


2 12 


50 
53 


1( 


)23 2( 


> 10 


4 43 


1( 


)2 26 


401 22 


74 


16 




6 


2 25 


2( 


)22 IE 


20 


4 38 


2( 


2 25 


35 1 21 


75 


15 




7 


2 36 


66 


3( 


)21 15 


30 


4 34 


3C 


2 24 


2C 


► 1 20 


76 


14 




8 


2 47 


59 


7 24 


4( 


)20 18 


40 


4 31 


4C 


2 23 


4C 


1 19 


77 


13 




9 


2 57 


62 


7 45 


6( 


(19 25 


50 


4 27 


6C 


2 21 


36 ( 


1 18 


78 


12 




10 


3 07 


65 


7 56 


2 C 


► 18 35 


12 


4 23 


22 


2 20 


3C 


1 17 


79 


11 




11 


3 16 


68 


8 07 


1C 


17 48 


10 


4 20 


10 


2 19 


37 C 


1 16 


80 


10 




12 


3 25 


71 


8 18 


2C 


17 04 


20 


4 16 


20 


2 18 


3C 


1 14 


81 


9 




13 


3 33 


74 


8 28 


3C 


16 24 


30 


4 13 


30 


2 17 


38 C 


1 13 


82 


8 




14 


3 41 


77 


8 38 


4C 


15 45 


40 


4 09 


40 


2 16 


30 


1 11 


83 


7 




15 


3 49 


80 


8 48 


5C 


15 09 


50 


4 06 


50 


2 15 


39 


1 10 


34 


6 




16 


3 56 


83 


8 58 


3 C 


14 34 


13 


4 03 


23 


2 14 


30 


1 09 


85 


5 




17 


4 04 


86 


9 08 


10 


14 04 


10 


4 00 


10 


2 13 


40 


1 08 


86 


4 




18 


4 11 


89 


9 17 


20 


13 34 


20 


3 57 


20 


2 12 


30 


1 07 


87 


3 




19 


4 17 


92 


9 26 


30 


13 06 


30 


3 54 


30 


2 11 


41 


1 05 


88 


2 




20 
21 

22 


4 24 
4 31 


95 


9 36 


40 


12 40 


49 


3 51 


40|2 10 


30 


1 04 


89 


1 




98 


9 45 


50 


12 15 


50 


3 48 


50 


2 09 


42 


1 03 


90 







4 37 


101 


9 54 


4 


11 51 


14 


3 45 


24 


2 08 


30 


1 02 








23 


4 43 


104 


10 02 


10 


11 29 


10 


3 43 


10 


2 07 


43 


1 01 








24 


4 49 


107 


10 11 


20 


11 08 


20 


3 40 


20 


2 06 


30 


1 00 








25 


4 55 


110 


10 19 


30 


10 48 


30 


3 38 


30 


2 05 


44 


59 








26 

27 
28 


5 01 


113 


10 28 


40 


10 29 


40 


3 35 


40 


2 04 


80 


58 








5 07 
5 13 


116 
119 


10 36 
10 44 


50 


10 11 


50 


3 33 


50 


2 03 


45 


57 








29 


5 18 


122 


10 52 


5 


9 54 


15 


3 30 


25 


2 02 


30 


56 








30 


5 24 


125 


11 00 


10 


9 38 


10 


3 28 


10 


2 01 


46 


55 








31 


5 29 


128 


11 08 


20 


9 23 


20 


3 26 


20 


2 00 


30 


54 








32 


5 34 


131 


11 16 


30 


9 08 


30 


3 24 


30 


1 59 


47 


53 








33 


5 39 


134 


11 24 


40 


8 54 


40 


3 21 


40 


1 58 


30 


52 








34 


5 44 


137 


11 31 


50 


8 41 


50 


3 19 


50 


1 57 


48 


51 








35 [ 5 49| 


1401 


11 39 


6 
10 
20 


8 28 
8 15 
8 03 


16 
10 

20 


3 17 
3 15 
3 12 


26 
10 
20 


1 56 
1 55 
1 55 


30 

49 

30 


50 
49 
49 
















TABLE VI. 


30 


7 15 


30 


3 10 


30 


1 54 


50 


48 








Dip of the Sea Horizon at 


40 


7 40 


40 


3 08 


40 


1 53 


30 


47 








different Distances from it. 


50 
7 


7 30 
7 20 


50 
17 


3 06 

3 04 


50 
27 


1 52 
I 51 


51 
30 


46 
D 45 










10 

20 


7 11 

7 02 
6 53 


10 
20 
30 


3 03 
3 01 
2 59 


15 
30 


I 50 
I 49 
L 48 


52 
30 

53 


» 44 
3 44 
) 43 








Dist. 


Hight of Eye in Ft. 




in 
Miles. 


5 


10 


15 


2( 


)J25 


30 


30 
40 


45 










T 


~7~ 


~ 


"7 


\~T 


~ 


6 45 


40 5 


I 57 


28 


L 47 


30 


) 42 








i 


11 


22 


34 


45 


56 


68 


50 


6 37 


50$ 


I 55 


15 


L 46 


54 0( 


) 41 








I 


6 


11 


17 


22 


28 


34 


8 


6 29 


18 5 


I 54 


301 


L 45 


55 0( 


) 40 








1 


4 


8 


12 


15 


19 


23 


10 


6 22 


10 5 


I 52 


45] 


L 44 


56 0( 


) 38 








1 


4 


6 


9 


12 


15 


17 


20 


6 15 


20 5 


J 51 


29 0] 


42 


57 0( 


) 37 








li 


3 


5 


7 


9 


12 


14 


30 


6 08 


30 5 


\ 49 


201 


41 


58 0( 


) 35 








U 


3 


4 


6 


8 


9 


12 


40 


6 01 


40 5 


! 47 


401 


40 


59 0( 


) 34 








2 


2 


3 


5 


6 


8 


10 


50 


5 55 


50 5 


! 46 . 


JO 01 


38 ( 


30 0( 


) 33 








21 


2 


3 


5 


6 


7 


8 


9 


5 98 


19 2 


44 


201 


37 ( 


51 0( 


) 32 








3 


2 


3 


4 


5 


6 


7 


10 


5 42 


10 2 


43 


401 


36 K 


32 0C 


1 30 








3* 


2 


3 


4 


5 


6 


6 


20 


5 46 


20 2 


41 : 


51 01 


35 ( 


33 0( 


> 29 








4 


2 


3 


4 


4 


5 6 


30 


5 41 


30 2 


40 


201 


33 ( 


34 0C 


28 








5 


2 


3 


4 


4 


5 5 


40 


5 25 


40 2 


38 


401 


32 ( 


55 0C 


26 




1 




6 


2 


3 


4 


4 


5| 5 


50 


5 20j 


50 2 37 i 


12 01 


31 \( 


i6 0C 


25 




I 




1 





WN 



JU 



-0 



fitt 



